THE THEORY OF MODULES OF SEPARABLY CLOSED FIELDS 2

PILAR DELLUNDE^{1}, FRANÇOISE DELON, AND FRANÇOISE POINT^{2}

Abstract. In [8], we determined the complete theory T_{e} of modules of separably closed fields of characteristic p and imperfection degree e, e ∈ ω ∪ {∞}. Here, for 0 ≠ e ∈ ω, we describe the closed set of the Ziegler spectrum corresponding to T_{e}. Further, we establish a correspondence between certain submodules and n-types and we investigate several notions of dimensions and their relationships with the Lascar rank. Finally, we show that T_{e} has uniform p.p. elimination of imaginaries and deduce uniform weak elimination of imaginaries.

MSC: 03C60; 03C95; 16D10; 12L12; 16S36

Keywords: separably closed fields; modules; skew polynomial rings; Ziegler spectrum; types; dimensions

1. Introduction

Let SCF_{p,e} be the theory of separably closed fields of characteristic p and of imperfection degree e, e ∈ ω. Let α be the Frobenius map and B a set of cardinality e. In [8], we considered the models of SCF_{p,e} as modules over the skew polynomial ring 𝔽_{p}(B)[t; α] (see [6]). It is straightforward to adapt our preceding results to a slightly more general setting, namely instead of 𝔽_{p}(B)[t; α], we will consider the skew polynomial ring R := K[t, α], where K is any field of characteristic p and imperfection degree e. We axiomatized the theory T_{e} of modules over R of the models of SCF_{p,e} and we showed that it admits quantifier elimination in the usual language of R-modules augmented with unary functions which are the analog in this context of the p-component functions. (In that article, we considered and treated the case of infinite imperfection degree analogously to the case of finite imperfection degree. Here, however, we will mostly restrict ourselves to the case of finite imperfection degree).

Our purpose here is two-fold, first to describe a closed subset of the Ziegler spectrum of R corresponding to the additive reduct we are considering, of the theory of the separable closure of K and second to show that in our particular setting one obtains tighter correspondence between the pure model theoretic objects as types, forking, ranks and their algebraic counterparts than in the field case.

Date: February 12, 2004.

1This work is partially supported by Spanish DGICYT grant PB98-1231.

2Senior Research Associate at the ”Fonds National de la Recherche Scientifique”.

2 PILAR DELLUNDE^{1}, FRANÇOISE DELON, AND FRANÇOISE POINT^{2}

We will first show that the theory T_{e}, in the case e ≠ 0, is not superstable (as in the field case). We will give a description of the closed set 𝕌_{K^{sep}} of the Ziegler spectrum of R with the additional hypothesis of countability on K. We will show on one hand that there are uncountably many torsion-free points (which are all elementarily equivalent from our previous analysis) and on the other hand that there are ℵ_{0} many non isomorphic points with non trivial torsion.

Then we will establish a correspondence between 1-types and certain submodules of the free R-module on countably many generators. We will prove an analogue of the Nullstellensatz (see p. 154 in [7]). We will use it together with the fact that the set of separable elements in our skew polynomial ring is a right denominator set, which entails that we have a separable closure for torsion-free summands of models of T_{e} and so a prime model construction, to show that one can read off the U-rank of a type in the length of a chain of sub-models.

We will show that definable subgroups in the models of T_{e} are (definably) connectedby-finite (this has already been proven in the case of separably closed fields (see [3]), but the proof there is more involved, as one can expect) and that a definable group over the empty set is definable by a positive primitive formula (this is unknown in general).

We will compare different notions of dependence relations, one (the R-dependence relation) coming from the fact that our ring is right Ore and the others coming from stability theory and the model theory of modules.

We will observe that one obtains with the rank coming from the R-dependence relation and the U-rank similar pathologies to the ones shown in [5], [3] and [1].

Finally, reading the results of Kucera and Prest in [15] in our context, we show that T_{e} has uniform p.p. elimination of imaginaries and from that we deduce uniform weak elimination of imaginaries, for e ∈ ω ∪ {∞}. Note that in the field case to get elimination of imaginaries, when e is finite, one has to add constants for elements of a p-basis and that one does not know, when e is infinite, an appropriate language to get elimination of imaginaries (see [7]).

2. Preliminaries

Let us recall the notations and basic results obtained in [8].

Let K be a field of characteristic p and imperfection degree e (which we do not assume to be separably closed) and let R be the skew polynomial ring K[t; α], with the commutation rule k.t = t.k^{α}, for all k ∈ K, where α is in this case the Frobenius map. The elements of R are of the form q(t) = ∑_{i=0}^{n} t^{i}.a_{i}, a_{i} ∈ K. The element q(t), or simply q, is called separable if a_{0} is non zero (and monic if a_{n} is 1). This ring R is right principal (see [6], chapter 2), but not left Ore whenever α is not surjective. The set X of all separable elements of R is a right denominator set, which means that, for all r ∈ R and x ∈ X, the set r.X ∩ x.R is not empty (see section 4, Definition 4.1).

Let B = {b_{i}; 0 ≤ i < e} be a p-basis of K with the convention that if e = 0, then B = ∅.

The language of right R-modules is L_{R} = {+, −, 0, r(.); r ∈ R}, where for any r ∈ R, r(x) := x · r (scalar multiplication by the ring element r), where x belongs to a right R-module. Let T_{R} be the theory of all right R-modules in this language.

We will always deal with right modules, and so we will no longer specify it.

Definition 2.1. Let e be finite. Given i ∈ p^{e}, let m_{i} be the monomial b_{0}^{i(0)}·. . .·b_{e−1}^{i(e−1)}. Let T (e) be the following L_{R}-theory:

(1) T_{R} the theory of all (right) R-modules,

(2) ∀x∃y, x = y · q(t), for any q(t) ∈ X,

(3) (∃x_{i})_{i=1}^{d} (the x_{i} ’s are linearly independent over 𝔽_{p} ∧ ⋀ x_{i} · q(t) = 0) and ¬[(∃x_{i})_{i=1}^{d+1} (the x_{i} ’s are linearly independent over 𝔽_{p} ∧ ⋀ x_{i} · q(t) = 0)], for any q(t) ∈ X monic of degree d,

(4) ∀x ∃(x_{i})_{i∈p^{e}} x = ∑_{i∈p^{e}} x_{i} · t · m_{i},

(5) ∀x∀(x_{i})_{i∈p^{e}}(∑_{i∈p^{e}} x_{i} · t · m_{i} = 0 → ⋀_{i∈p^{e}} x_{i} = 0).

It is convenient to single out different sub-theories of T (e), namely, let T be the subtheory where we do not specify the torsion, so T consists of axioms schemes 1, 2, 4 and 5; let T_{λ} be axioms schemes 1, 4 and 5 and finally let T_{sep} be axioms schemes 1, 2 and 3. Let T^{tf} be the theory T plus the scheme of axioms (one for each r ∈ R −{0}): ∀m (m · r = 0 → m = 0). (We will use this superscript ^{tf} to denote such extension of a theory of R-modules.) Note that in any model M of T_{λ} one can define p^{e} unary additive functions λ_{i}, i ∈ p^{e}, as follows. By axiom 4, any element x of M can be written as x = ∑_{i∈p^{e}} x_{i} · t · m_{i} and we set λ_{i}(x) = x_{i}; note that axiom 5 ensures that it is well-defined.

Each of the class of models of either T_{λ}, T or T^{tf} is closed under direct summands and direct products. Note also that if M is a model of T^{tf} and N be a model of T (e), then M ⊕ N is a model of T (e).

In [8] we considered only the case where K = 𝔽_{p}(B), but it is straightforward to extend our former results to the present setting. Let us rephrase them as follows. In order to get a quantifier elimination result, we considered an expansion by definition of T (e), which will be denoted by T_{e}. We extended the usual R-module language by adding new unary additive functions (the functions λ_{i}’s defined above). We will repeatedly use the fact that they are definable by primitive positive (p.p.) L_{R}- formulas.

Let L = L_{R} ∪ {λ_{i}; i ∈ p^{e}}, where the λ_{i}’s are unary functions.

Let T_{e} (respectively T_{free}) be the following L-theory:

(1) T (e) (respectively T),

(2) ∀x x = ∑_{i∈p^{e}} λ_{i}(x) · t · m_{i}.

It is straightforward to show that any model M of T can be uniquely expanded to a model of T_{free} and any pure submodule of M can be expanded to an L-substructure of M.

In [8] Corollary 6.5, we showed that in T_{free}, any p.p. L_{R}-formula is equivalent to a conjunction of atomic L-formulas. So, conversely, any L-substructure N of a model M of T_{free} is a pure submodule of M.

We will use several times the following fact (proof of 3.5 in [8], item 3): let N be an L-substructure of a model M of T_{free}, let x ∈ M be such that x · q = n ∈ N for a separable q ∈ R. Then λ_{i}(x), i ∈ p^{e}, belongs to N⟨x⟩_{R}, the R-submodule of M generated by N and x. More precisely, we may assume without loss of generality that q = 1 − t.q′, so t.q′ = ∑_{i} \sqrt[p]{q′_{(i)}}.t.m_{i}, and then λ_{i}(x) = x · \sqrt[p]{q′_{(i)}} + λ_{i}(n).

Together with the relative quantifier elimination result mentioned above, this implies that the torsion submodule M_{tor} of any model M of T_{free} is an L-substructure and so a pure submodule (see [8] Corollary 6.6).

The index of two p.p. definable subgroups with one included in the other is either 1 or infinite simultaneously in all torsion-free summand of a model of T_{e} (see Lemma 6.8 in [8]). Note that it implies that T^{tf} is complete.

Furthermore, it entails that T_{e} (respectively T_{free}^{tf}) admits quantifier elimination, even positive quantifier elimination i.e. any positive primitive formula is equivalent to a conjunction of atomic formulas (6.5 and remark 6 in [8]). Moreover, using axiom 5 of T (e), and iterates of the λ functions (see Notation 3.3 in [8]), it follows that a conjunction of atomic formulas is equivalent in T_{free} to one atomic formula: ⋀_{k<p^{m.e}} t_{k}(x̄) = 0 ↔ ∑_{k∈p^{m.e}} t_{k}(x̄) · t^{m} · m_{k} = 0.

We showed that in a model of T_{e} the torsion submodule is isomorphic to the separable closure K^{sep} of the field K and in addition that any L-substructure M is contained in a minimal submodel M^{sep} of T_{e}, the separable closure of M (Corollary 5.9 in [8]).

Therefore, any pure-injective model of T_{e} can be decomposed as a direct sum of the pure-injective hull of K^{sep} and a torsion-free summand. It follows that T_{e} is a complete reduct of the theory of the separably closed fields of characteristic p and imperfection degree e and so is decidable. This entails that the L_{R}-reduct T (e) of T_{e} is complete and decidable as well.

Considering now two models N and N′ of T such that N_{tor} ≅ N′_{tor}, note that N ≡ N′. Indeed, by the same argument as above but now applied to N_{tor} (instead of K^{sep}), we have that N_{tor} is pure in N and so if we denote by N^{∗} a non principal ultrapower of N, we get on one hand that N^{∗} = N_{tor}^{∗} ⊕N_{0} and on the other hand that N′^{∗} = N′_{tor}^{∗} ⊕ N_{1}, where N_{0} ⊆ N^{∗}, N_{1} ⊆ N′^{∗} and N_{0}, N_{1} ⊧ T^{tf}. Since N_{0} ≡ N_{1}, we obtain N′ ≡ N.

For the axiomatisation in the infinite imperfection degree case (namely when e is infinite), see [8] sections 4 and 7. We got analogous results to the ones in the finite imperfection degree case. Recall that we have denoted the corresponding theories by T_{∞}. These last theories will come up in the last section when we discuss the elimination of imaginaries.

3. Ziegler space of indecomposables

Let Zg_{R} be the Ziegler spectrum associated with the ring R. It is a topological space whose points are the indecomposable pure-injective R-modules and whose basic open sets are of the form (ϕ/ψ) where ϕ, ψ are two p.p. formulas in one variable with ψ implying ϕ in T_{R}, and (ϕ/ψ) consists of all indecomposable pure-injective R-modules where the index of ψ in ϕ is strictly greater than 1 (see [22]).

Definition 3.1. (See [22], Section 8 and Corollary 4.10). Let M be an R-module. Let 𝕌_{M} be the class of all indecomposable pure-injective modules which are direct summands in modules elementarily equivalent to M. Such subsets are the closed subsets of Zg_{R}.

Since T (e) is complete, any model of T (e) is elementarily equivalent to K^{sep}. Therefore, the pure-injective indecomposable R-modules occurring as direct summands of models of T in Zg_{R} belong to 𝕌_{K^{sep}}.

Since the theory T (e), e ∈ ω is decidable, this closed subset 𝕌_{K^{sep}} of Zg_{R} should be well understood (see Remark after Theorem 9.4 [22]). Note that a pure-injective indecomposable module which is not in 𝕌_{K^{sep}} can be obtained as follows. Consider R̂ the completion of R i.e. it consists of the power series of the form ∑_{i∈ℕ} t^{i}.a_{i}, then consider the right denominator set {t^{i}; i ∈ ℕ} and take the field of fractions of R̂ with respect to that subset. One obtains a skew field (see [6] section 2.4) whose elements are of the form ∑_{i∈ℕ} t^{i}.a_{i}.t^{−n}, n ∈ ℕ. It is not a model of T (e) because it does not satisfy axiom 5. (Take a ∈ K − K^{p}, then a = a.t^{−1}.t = ∑ a_{i}^{p}.t^{−1}.t.m_{i}.)

First, we prove, as in the case of separably closed fields (see [21]), that the theory T (e) is not superstable, for e ∈ (ω − {0}) ∪ {∞}.

This follows from the following more general result. In the next proposition and lemma, we place ourselves in the following setting, namely in a class of modules over a skew polynomial ring S := F [t; α], where F is an infinite field, α an endomorphism of F, and where multiplication by t is injective.

Proposition 3.2. Let M be an S-module. Assume in addition that α is a non surjective endomorphism of F and that in M multiplication by t is not surjective. Then, the theory of M is stable but not superstable.

Proof: First, any complete theory of modules is stable (see [18] Theorem 3.1). Second, we apply Theorem 2.1 in [22]. Let us show that in the following chain of p.p. definable subgroups M ⊇ M · t ⊇ M · t^{2} ⊇ ⋯ each one is of infinite index in its predecessor. By the way of contradiction, suppose that [M : M · t] = ℓ (the argument is the same for any consecutive pair in the chain); let f_{0}, ⋯, f_{ℓ} be distinct elements of F − {0} and let m ∈ M − M · t. Then, the elements m · f_{0}^{α}, m · f_{1}^{α}, ⋯, m · f_{ℓ}^{α} are in different classes modulo M · t. For if not, then m · (f_{i}^{α} − f_{j}^{α}) ∈ M · t, where 0 ≤ i < j ≤ ℓ i.e. m · (f_{i}^{α} − f_{j}^{α}) = n · t, for some n ∈ M. So, m = n · t.((f_{i} − f_{j})^{α})^{−1} = n · (f_{i} − f_{j})^{−1}.t by the commutation rule and so m ∈ M · t, a contradiction. □

To apply it to T (e), it suffices to take M := K^{sep}.

Then, we note the following property of models of T (which we prove in a slightly more general setting).

Lemma 3.3. Let N be an S-module where multiplication by t is injective. Then N is a fully faithful S-module.

Proof: Recall that N is fully faithful if for any non-zero S-submodule M of N, ann(M) := {r ∈ S; ∀m ∈ M, m · r = 0} is equal to the zero ideal (see [11] p.31).

First, note that it is a two-sided ideal of S. Since the right ideals are principal, ann(M) is of the form f.S with in this case the additional property that S.f ⊆ f.S. Whenever f is non zero, such an element is called right invariant (see [6] p.57) and so f is of the form t^{n}.c, where c ∈ F − {0} (see Proposition 2.2.8 in [6]). Therefore, if M is a non zero module, ann(M) = {0}. □

In the following, assuming that e ∈ ω − {0} and that K is countable, we will show that if M is a model of T^{tf}, then there are 2^{ℵ_{0}} points in 𝕌_{M} ⊆ 𝕌_{K^{sep}} We will need the following technical lemma (which holds whatever the cardinality of K is).

Lemma 3.4. Let e ∈ ω − {0} and ϕ, ψ be a pair of p.p. L_{R}-definable subgroups of M with T_{R} ⊧ ψ → ϕ and T^{tf} ⊧ [ϕ : ψ] > 1. Then, there exists a p.p. L_{R}-formula γ with T_{λ} ⊧ ψ → γ → ϕ such that the indices of ψ in γ and of γ in ϕ are infinite in any model of T^{tf}. Further there is another p.p. L_{R}-formula γ′ with the same properties and such that, modulo T^{tf}, γ and γ′ are incomparable in the lattice of p.p. L_{R}-definable subgroups.

This means that, if we consider the pair (ϕ, ψ) as a sublattice of the lattice of p.p. definable subgroups in models of any completion of T, its m-dimension and its breadth, or its width, in the sense of [18] p.205, are undefined (for the fact that the breadth of a pair is undefined iff its width is undefined, see [18] Lemma 10.7). Proof: Recall that any model of T can be expanded to an L-structure and so becomes a model of T_{free}. Since T_{free} admits positive quantifier elimination, ϕ(x) and ψ(x) are equivalent (in T_{free}) to a conjunction of atomic L-formulas. Using Corollary 3.3 in [8] on normal form for terms, as in Lemma 6.8 in [8], substituting new variables x_{i} for the unary functions λ_{i}(x), we get that ϕ(x) (respectively ψ(x)) is equivalent in T to an L_{R}-formula of the form ∃x̄ θ(x, x̄) (respectively ∃x̄ θ′(x, x̄)), where θ(x, x̄) is the formula x = ∑_{i∈p^{en}} x_{i} · t.m_{i} ∧ x̄ · P.C = ¯0, (respectively θ′(x, x̄) is the formula x = ∑_{i∈p^{en}} x_{i} · t.m_{i} ∧ x̄ · B = ¯0), where n ≥ 0, P is a permutation matrix and C is a lower triangular p^{en} × (j − 1)-matrix with coefficients in R, and with only separable coefficients on its diagonal. Since |ϕ(x)| is infinite, one has j ≤ p^{en}. Set z̄ = x̄ · P. Then for any solution (a, ā) of θ in a model of T, the tuple ā′ := ā·P has the property that (a′_{0}, ⋯, a′_{j−2}) ∈ ⟨a′_{j−1}, ⋯, a′_{p^{en}−1}⟩_{R}^{sep}, with the convention that if j = 1 (namely if ϕ(x) is equivalent to x = x) the tuple (a′_{0}, ⋯, a′_{j−2}) is empty.

Since ϕ ∧ ψ is equivalent to ψ in T_{R}, using Euclidean division in R and Gauss elimination as in [8] 6.1 (first between z̄ · C = ¯0 and z̄ · P^{−1}.B = ¯0 and then inside what is, after having done that, the block of last columns of the same breadth as B), we obtain a permutation σ on p^{en} which is the identity on {0, ⋯, j − 2} if j ≠ 1, and an invertible matrix Q with coefficients in R such that, if P′ is the matrix of σ, then P′.P^{−1}.B.Q =: B′ =: (b′_{i,h}) is lower triangular and has the property that the coefficients b′_{i,i}, 0 ≤ i ≤ j − 1, are non zero. So, z̄ · P^{−1}.B = ¯0 is equivalent to the system z̄ · P^{′−1}.B′ = ¯0; set z̄′= z̄ · P′^{−1} = (z_{0}, ⋯, z_{j−2}, z_{σ(j−1)}, ⋯, z_{σ(p^{en}−1)}), with the convention that if j = 1 then the first part of the tuple is empty. Moreover, since there exists x in a model of T^{tf} such that ϕ(x) ∧ ¬ψ(x), we can take σ such that the coefficient b′_{j,j} in the matrix B′ is non zero. Consider the equation ∑_{k=j−1}^{p^{en}−1} z′_{k} ·b′_{k+1,j} = 0. Applying Lemma 3.4 in [8], we may always assume that at least one coefficient b′_{k_{0},j}, for some k_{0} ∈ {j, ⋯, p^{en} −1}, is separable. (Otherwise, we replace the equation by the system ⋀_{i∈p^{e}} ∑_{k=j−1}^{p^{en}−1} z′_{k} · \sqrt[p]{b′′_{k+1,j}_{(i)}} = 0, where each b′_{k+1,j}=t.b′′_{k+1,j}.)

The equation ∑_{k=j−1}^{p^{en}−1} z′_{k} · b′_{k+1,j} = 0 is equivalent to the conjunction ⋀_{i∈p^{e}} λ_{i}(∑_{k=j−1}^{p^{en}−1} z′_{k} · b′_{k+1,j}) = 0. Now, the basic idea for defining the intermediate subgroup is to intersect ϕ with one of the elements of the above conjunct. Since the functions λ_{i} are additive, it is equivalent to ⋀_{i∈p^{e}} ∑_{k=j−1}^{p^{en}−1} λ_{i}(z′_{k} ·b′_{k+1,j}) = 0. We write b′_{k_{0}+1,j} as α + t.q, where α belongs to K − {0} and q to R. We may assume without loss of generality that α = 1. We have λ_{i}(z′_{k_{0}} · b′_{k_{0}+1,j})=λ_{i}(z′_{k_{0}}) +λ_{i}(z′_{k_{0}} · t.q) and, by Remark 2, item 3 in [8], we have λ_{i}(z′_{k_{0}} · t.q)=z′_{k_{0}} · \sqrt[p]{q_{(i)}}. Replacing in that last term, z′_{k_{0}} by ∑_{h∈p^{e}} λ_{h}(z′_{k_{0}}) · t.m_{h}, we get a system of equations ⋀_{i∈p^{e}} γ_{i}(\overline{λ(z′_{k})}) where the formula γ_{i}(\overline{λ(z′_{k})}) is

λ_{i}(z′_{k_{0}}) · (1 + t.m_{i}.\sqrt[p]{q_{(i)}}) + ∑\begin{matrix}h∈p^{e} \\ h≠i \end{matrix} λ_{h}(z′_{k_{0}}) · t.m_{h}.\sqrt[p]{q_{(i)}} + ∑p^{en}−1 ; \begin{matrix}k=j−1 \\ k≠k_{0} \end{matrix} λ_{i}(z′_{k} · b′_{k+1,j}) = 0.

For i ≠ h, the coefficients of λ_{i}(z′_{k_{0}}) in γ_{i}, respectively in γ_{h}, are distinct, namely that in γ_{i} is separable and that in γ_{h} is not. To see now that γ_{i} ∧ γ_{h} ∧ ϕ(x) is strictly smaller than say γ_{i} ∧ ϕ(x) (the argument for γ_{h} is similar), we proceed as follows. In order to simplify the notation, set v_{1} := λ_{i}(z′_{k_{0}}) and v_{2} := λ_{h}(z′_{k_{0}}) and write γ_{i} as v_{1} · q_{1} + v_{2} · p_{1} + u = 0, where q_{1} ∈ X (the set of separable elements of R), p_{1} ∈ R − X and u is a term in the remaining variables; similarly write γ_{h} as v_{1} ·p_{2} +v_{2} ·q_{2} +w = 0, where q_{2} ∈ X, p_{2} ∈ R − X and w is a term in the other variables.

Now, X is a right denominator set and so ∃t_{1}, t_{2} ∈ R with q_{1}.t_{1} = p_{2}.t_{2} and t_{2} ∈ X. So, the system consisting of the two equations γ_{i} ∧ γ_{h} is equivalent in T^{tf} to the system

\{ \begin{matrix}v_{1} · q_{1} & + & v_{2} · p_{1} & + & u = 0 \\ & & v_{2} · (p_{1}.t_{1} − q_{2}.t_{2}) & + & u · t_{1} − w · t_{2} = 0 \end{matrix}

with (p_{1}.t_{1} − q_{2}.t_{2}) ∈ X. Now, take u and w arbitrarily, choose v_{2} such that it does not satisfy the second equation of the system (this is possible since there is only one such element whenever u and w are fixed) and then choose v_{1} in order to satisfy γ_{i}: this is always possible since its coefficient is separable in that equation. Then choose z′_{k} for k < (j − 1) in order to satisfy ϕ, this is always possible since we are in a model of T. We finish the proof by applying Lemma 6.8 of [8]. □

Corollary 3.5. Suppose that e ∈ ω − {0} and K is countable and let M be a model of T^{tf}. Then there are 2^{ℵ_{0}} points in 𝕌_{M} (and therefore in 𝕌_{K^{sep}}).

Proof: We apply Lemma 8.3 in [22] (it is there where the countability hypothesis is needed). □

Corollary 3.6. Suppose that e ∈ ω − {0} and K is countable. Then, there exists a model N of T^{tf} which has no indecomposable direct summand.

Proof: A complete theory of modules over a countable ring has a continuous part zero iff its width (or equivalently its breadth) is defined (M. Ziegler uses the term bounded )([22] Theorem 7.1). □

Corollary 3.7. Suppose that e ∈ ω − {0} and K is countable. Then, the Cantor-Bendixon rank of the closed subset corresponding to T^{tf} of the Ziegler spectrum of R is undefined. □

Definition 3.8. (See [22] before Theorem 3.6). Let A be a subset of a pure-injective R-module M. The hull H(A) of A (in M) is a maximal small extension of A ([22] Corollary 3.10) i.e. every partial homomorphism from A to an R-module whose restriction to A is a partial isomorphism (recall that a partial isomorphism is a partial mapping that preserves p.p. formulas and negations of p.p. formulas), and it is defined by the following two properties:

(i) H(A) is a pure-injective module such that A ⊆ H(A) ⊆ M, and H(A) is pure in M. (ii) for any pure-injective module B with A ⊆ B ⊆ H(A) and B pure in M, then B=H(A).

The injective hull E(A) of a subset A in an injective R-module M is defined similarly as above replacing “pure-injective” by “injective” and “pure” by “included”.

Notation 3.9. (See Notation after Theorem 3.6 in [22]). Let M be a pure-injective R-module, a an element of M and p its p.p.-complete type over ∅ (i.e. the set of p.p. formulas and negations of such satisfied by a in M). A consequence of the Baur-Monk quantifier elimination result is that, in any theory of modules, the type of an element is axiomatized by its p.p.-complete type. The hull H(a) of a in M is up to isomorphism determined by its p.p.-complete type p and so we will write indifferently H(p) or H(a).

Existence and uniqueness of the hull was proven by Fisher and may be found in [22] Theorem 3.6 or in [18] chapter 4.1. In particular, let A ⊆ M, M′, where M, M′ are pure-injective modules and the p.p. type of A is the same in M and in M′. Then, H_{M}(A) ≅ H_{M′}(A). The hull of any model M of T (e) (respectively T^{tf}) can be expanded to an L-structure (the functions λ_{i} are p.p. definable) and this expansion is a model of T_{e} (respectively T_{free}^{tf}) (both theories are model-complete). Note that hull and pure hull coincide in this case (for a definition of pure hull, see [22] p.162).

Further, suppose that N is an R-module included in a model M of T_{e}. Then, H(N) contains the L-substructure of M generated by N, in particular if N is a pureinjective R-submodule, it can be expanded to an L-structure. Indeed, the functions λ_{i} are p.p. definable and so for a ∈ N, the set Λ := {λ_{i}(a) : i ∈ p^{e}} is small over N and therefore is included in H(N).

Corollary 3.10. Assume K countable and that e ∉ {0, ∞}, and let S_{1}(∅) be the set of complete 1-types over ∅ of T^{tf}. Then there are 2^{ℵ_{0}}-points in S_{1}(∅). □

Now, we will consider the points N in 𝕌_{K^{sep}} for which the torsion submodule N_{tor} is non-trivial. We already know that N_{tor} ⊆ K^{sep} (see [8]) and so N is included in H(K^{sep}).

Recall that q ∈ R is prime separable if it is separable, non invertible, and cannot be written as a product of two non zero invertible elements of R.

Lemma 3.11. Let N be a model of T which is pure-injective and indecomposable and suppose that N_{tor} contains a non zero element a annihilating some prime separable element p_{0} of R. Let N′ be another model of T which is pure-injective and indecomposable and suppose that ann(p_{0}) ∩ N′ is non-trivial, i.e. of cardinality strictly bigger than 1. Then N ≅ N′.

Proof: First let us note that the type of a is determined by the fact that a · p_{0} = 0 ∧ a ≠ 0. First, since the element p_{0} is prime, it is of minimal degree such that it annihilates a and so any polynomial q annihilating a is right divisible by p_{0}. Then, any p.p. L_{R}-formula is equivalent to an atomic L-formula ([8] Corollary 6.5) and the L-substructure generated by a coincides with the R-submodule generated by a. So any formula satisfied by a is equivalent to a boolean combination of atomic formulas of the form a · q = 0, for some q ∈ R and whether the later holds or not only depends on whether p_{0} divides q. □

The preceding Lemma shows that there are at most max{|K|, ℵ_{0}} non isomorphic indecomposable direct summands in H(K^{sep}). When K is countable, we will show that there are exactly ℵ_{0} of them.

First, we need to investigate in more detail the structure of those elements of 𝕌_{K^{sep}} with non-trivial torsion.

We begin by proving a property of pure-injective indecomposable models of T_{free}, which follows from the fact that T_{free} has positive quantifier-free elimination. We need the following notion which appears in the setting of R-modules (see for instance [11] p.71).

Definition 3.12. An L-structure is trivial if its domain is equal to {0}. An Lstructure is uniform if the intersection of any two non-trivial L-substructures is nontrivial.

Lemma 3.13. Let N be a model of T_{free} which is pure-injective and indecomposable. Then N is uniform.

Proof: In a pure-injective indecomposable R-module, any two non-zero elements a, b are linked (see Corollary 4.11 in [18]) i.e. there is a p.p. formula ϕ(x, y) such that ϕ(a, b) and ¬ϕ(a, 0). Then, as in Corollary 16.7 in [18], one applies the fact that T_{free} has positive quantifier elimination. □

Lemma 3.14. Let N be a model of T which is pure-injective and indecomposable and suppose that N_{tor} contains a non zero element a annihilating some prime separable element p_{0} of R. Then, for any other element b of N_{tor} − {0}, there exists s ∈ R with a = b · s. Moreover, a prime separable element q of R is annihilated by an element of N − {0} iff it has the same degree as p_{0} and if there exist q_{1}, q_{2} ∈ R with deg(q_{1}), deg(q_{2}) < deg(p_{0}) such that q.q_{1} = q_{2}.p_{0}. In addition, the annihilators of two prime separable elements of R, whenever they are non-trivial, are isomorphic as 𝔽_{p}-vector spaces.

Proof: We first expand N into an L-structure, this expansion is a model of T_{free} and so by the preceding lemma, there exist L-terms σ, τ such that σ(a) = τ(b) ≠ 0. Since a, b both belong to N_{tor}, we have that, for i ∈ p^{e}, λ_{i}(a) (respectively λ_{i}(b)) belongs to the R-submodule generated by a (respectively by b). So we have a relation of the form (*) a · r + b · s = 0, with r, s ∈ R and a · r ≠ 0. Assume now that r is of minimal degree but does not belong to K, w.l.o.g., deg(r) < deg(p_{0}). Then, applying the right Euclidean algorithm in R, we get: p_{0} = r.r_{1} + r_{2} with deg(r_{2}) < deg(r) and r_{2} ≠ 0 (p_{0} is prime), hence a · r_{2} + b · s.r_{1} = 0, a contradiction. So, we may assume that the relation (*) is of the form a = b · s′, with deg(s′) less than deg(q) where q is of minimal degree such that b · q=0. (Note that such q is necessarily separable.)

Now, assume furthermore that q is a prime polynomial. Since b · s′.p_{0} = 0, there exists q_{1} in R such that q.q_{1} = s′.p_{0}. Note that since deg(q) > deg(s′), s′ and q are relatively prime. Now we apply Theorem 11 in [16], so p_{0} is divisible by s^{′−1}.lcm(s′, q) and deg(s^{′−1}.lcm(s′, q))=deg(q). But, p_{0} is prime, so p_{0} = s^{′−1}.lcm(s′, q) and deg(p_{0})=deg(q). We also have that whenever v ∈ ker(q), v · s′ ∈ ker(p_{0}), and since deg(s′) < deg(q), ker(s′) = {0}. So, there is an injection from ann_{N}(q) into ann_{N}(p_{0}), both finite. But in this reasoning the roles of a and b can be reversed and so there is a bijection between the two annihilators. The condition on the prime separable q is that it has the same degree as p_{0} and that there exist q_{1}, q_{2} with deg(q_{1}), deg(q_{2}) < deg(p_{0}) such that q.q_{1} = q_{2}.p_{0}. □

Proposition 3.15. Assume K is countable. Then, there are exactly ℵ_{0} non elementarily equivalent indecomposable pure-injective direct summands in H(K^{sep}).

Proof: Let {p_{n} : n ∈ ℕ} be an enumeration of the prime separable elements of R. Given n, let p_{n+k} be the first element on the list such that there is in K^{sep} a zero of p_{n+k} which does not belong to the 𝔽_{p}- subspace generated by the annihilators of the p_{i}, 1 ≤ i ≤ n. Consider the set of formulas q_{n}(x) = {¬(∃x_{1}, ⋯, x_{n} (x = ∑_{i=1}^{n} x_{i} ∧ ⋀_{i=1}^{n} x_{i}.p_{i} = 0)) ∧ x.p_{n+k} = 0}. Then q_{n}(x) satisfies the hypothesis of Lemma 4.7 in [22]. So, there is an extension t_{n} of q_{n} which is consistent with K^{sep}, complete and indecomposable, in the sense that H(t_{n}) is indecomposable. We get that H(t_{n}) embeds in H(K^{sep}). This provides at least ℵ_{0} non elementary equivalent indecomposable direct summands in H(K^{sep}). By Lemma 3.11, we have at most ℵ_{0} non isomorphic pure-injective indecomposable direct summands in H(K^{sep}). □

Question Does the conclusion of the above proposition holds without the hypothesis of countability?

In [8], we pointed out that any model of T_{λ} can be viewed as a module over the ring of endomorphisms R′ := R[λ_{i}; i ∈ p^{e}]. We will prove in 5.3 that R′ is the ring of all definable scalars R_{𝕌_{K^{sep}}} (see [4] p. 189). It is classical to consider modules as modules over their ring of definable endomorphisms and T. Blossier has developed in [1] this point of view for separably closed fields. He has also given an abstract description of R′.

4. Separable closure

In this section, we will show that we can embed any torsion-free R-module which is a model of T_{λ} into a model of T. Further, we will prove that any model of T^{tf} can be viewed as a module over a non commutative ℤ-valued valuation ring (see [6] p.83 2.6), namely the ring of fractions R.X^{−1} of R with respect to the set X of separable elements of R.

In the next section, we will use this construction in order to establish the correspondence between certain submodules of the free countably generated R-module and L_{R}-types over T (e).

Definition 4.1. A multiplicative subset X of a domain S is a right denominator set if ∀r ∈ S, ∀x ∈ X, r.X ∩ x.S ≠ ∅.

In such a situation there exists a ring of fractions S.X^{−1} for S with respect to X (we use here that two elements in a right denominator set have a right common multiple belonging to this set, see [11] Lemma 9.6 and Theorem 9.7). Then any right S-module A has a module of fractions A · X^{−1} with respect to X, which carries a unique right S.X^{−1}-module structure ([11], Theorem 9.13 and 9.12 (b)), in fact A · X^{−1} ≃ A ⊗_{S} (S.X^{−1}) and any element m of A ⊗ S.X^{−1} may be written as a ⊗ x^{−1} with a ∈ A and x ∈ X (see [11] Proposition 9.14 and its proof).

Note that A embeds in its module of fractions whenever A has no X-torsion i.e. {a ∈ A : ∃x ∈ X a · x = 0} = {0}.

Lemma 4.2. The set X of separable elements of the skew polynomial ring R is a right denominator set and R embeds canonically in R.X^{−1} . □

This is Lemma 5.7 in [8] and the fact that X contains 1 and that R has no Xtorsion.

Proposition 4.3. Let A be a torsion-free right R-module. Then, (1) A embeds in A · X^{−1} ⊕ K^{sep} which is a model of T_{sep}.

(2) If in addition A is a model of T_{λ}, then A · X^{−1} is a model of T^{tf}. Therefore, if A is a model of T^{tf}, then A can uniquely be endowed with a structure of right R.X^{−1}-module (compatible with its right R-module structure).

So, T_{free}^{tf} is a model-companion of the expansion by definition to L of T_{λ}^{tf}.

Proof: 1. By the above Lemma, X is a right denominator set. Since A is torsion-free, A embeds in A · X^{−1}. Consider now the R-module A · X^{−1} ⊕ K^{sep}. It satisfies axiom scheme 2 since A · X^{−1} is X-divisible and it satisfies axiom scheme 3 since A · X^{−1} is X-torsion-free (see Proposition 9.12 (a) in [11]).

2. Assume now in addition that A is a model of T_{λ}. We will show that A · X^{−1} can be expanded to an L-structure. By definition, for any element b of A · X^{−1}, there exists q ∈ X such that b · q ∈ A. Suppose that the degree of q is minimal such and denote by a the element b · q. Then, write q = α − t.q_{0}, where α ∈ K − {0}. W.l.o.g., we may assume that α = 1. We have b = a + b · t.q_{0}. So, we define λ_{i}(b) := λ_{i}(a) + b · \sqrt[p]{(q_{0})_{(i)}}, hence b = ∑_{i} λ_{i}(b) · t.m_{i} (see Notation 3.2 remark 2 and Proposition 3.5 (Proof of item 3) in [8]). It remains to show that if ∑_{i} b_{i} · t.m_{i} = 0, then ⋀_{i} b_{i} = 0. Using the isomorphism between A · X^{−1} and A ⊗ R · X^{−1}, we further may write each b_{i} as a_{i}⊗x_{i}^{−1}, with x_{i} ∈ X and a_{i} ∈ A. Since X is a right denominator set of R, there exists r_{i} ∈ R − {0} and x ∈ X such that r_{i}.x^{−1}=x_{i}^{−1}. So, we have ∑_{i} a_{i} · r_{i}.x^{−1}.t.m_{i}=0. Now, we view this sum as an element of the tensor product of the right R-module A · X_{R}^{−1} with the left R-module _{R}R. We have the isomorphism A · X^{−1} ⊗_{R} R ≅ A · X^{−1}. We consider the elements t.m_{i}’s as a set of left generators of _{R}t.R. There exist elements u_{j}, j ∈ J, of A · X^{−1} and elements a_{ij} in R such that (a_{ij}).(t.m_{i}) = 0 and (a_{i} · r_{i}.x^{−1}) = (u_{j}) · (a_{ij}) (see Proposition 8.8, Chapter 1 in [19]). But the ring R is a domain and so the matrix (a_{ij}) is the zero matrix. This implies that a_{i} · r_{i}.x^{−1}=0 for each i and, A being torsion-free, this entails that each a_{i} and so each b_{i} is equal to 0. The last part of the statement is clear. □

Remark 4.4. As in the commutative case (see [22] Lemma 5.3), we have the following. Given M, N two R.X^{−1}-modules and M_{R}, N_{R} their reducts in the language of R-modules, then, M ≡ N iff M_{R} ≡ N_{R}. (The verification is straightforward, using simply that, given n elements r_{1}.x_{1}^{−1}, ⋯, r_{n}.x_{n}^{−1} ∈ R.X^{−1} with r_{1}, ⋯, r_{n} ∈ R, x_{1}, ⋯, x_{n} ∈ X, there exist x ∈ X, s_{1}, ⋯, s_{n} ∈ R such that r_{1}.x_{1}^{−1} = s_{1}.x^{−1}, ⋯, r_{n}.x_{n}^{−1} = s_{n}.x^{−1}. Hence for ϕ a p.p. L_{R.X^{−1}}-formula, ϕ(z̄) := (∃ȳ ȳ · A = z̄), there is some x ∈ X such that ϕ · x := (∃ȳ ȳ· A.x = z̄) is a p.p. L_{R}-formula; now ϕ and ϕ · x are equivalent in any X-torsion-free and X-divisible R-module.)

Moreover, as in the commutative case, to describe the pure-injective indecomposable modules over R.X^{−1} is equivalent to describing the pure-injective indecomposable X-torsion-free modules over R.

From the preceding Proposition and the fact that a direct sum of a model of T (e) with a model of T^{tf} is a model of T (e), we deduce the following result:

Corollary 4.5. Let A be a model of T_{λ}^{tf}. Then A can be embedded in a model of T (e). □

In Proposition 5.8 [8], given a model M of T_{sep} and an R-submodule A of M, we constructed the separable closure of A in M, which we denoted by A^{sep}. Moreover, in case M is a model of T_{e}, A^{sep} is a model of T_{e}. Let us see the relationship between A^{sep} and the module of fractions A · X^{−1}. Assume that K^{sep} ⊆ A, then A^{sep}/K^{sep} is a module of fractions for A/K^{sep} and therefore by uniqueness of the R.X^{−1} module structure on A/K^{sep}, we have that A^{sep}/K^{sep}= (A/K^{sep}) · X^{−1}. Observe that A^{sep}/K^{sep} is also a module of fractions for A, so we get that

A · X^{−1} ≅ A^{sep}/K^{sep} ≅ (A/K^{sep}) · X^{−1}.

Let us denote by M the monster model of T (e) and by M_{L} the expansion of M to a model of T_{e}. Given any subset A of M, we denoted by (A)_{L} the L-substructure of M_{L} generated by A and ⟨⟨A⟩⟩ := ((A)_{L})^{sep} is the model-theoretic algebraic closure of A (see [8]), it is also the unique minimal prime model of T_{e} over A. We give now a characterization of the definable closure of A, which we denote by (A)_{def}.

Lemma 4.6. Let A be a subset of M, then (A)_{L} = (A)_{def} and thus every element of (A)_{def} is equal to an L-term in elements of A.

Proof: Clearly (A)_{L} ⊆ (A)_{def}. Suppose that there is an element x ∈ (A)_{def} − (A)_{L}. Since ((A)_{L})^{sep} is a model of T_{e} and T_{e} model-complete (see Corollary 5.10 in [8]), x ∈ ((A)_{L})^{sep}. So there is q ∈ R separable and of minimal degree with the property that x · q ∈ (A)_{L}. Among all such elements x, take one with the degree of the corresponding polynomial q minimal. Note then that such q is prime. Let a ∈ (A)_{L} be such that x · q = a. We claim that q ∈ K and so we get a contradiction. Assume, searching for a contradiction, that deg(q) ≥ 1, so there is y ∈ M, y ≠ x such that y · q = a. Since q is prime, the map ϕ : ⟨x⟩_{R} + (A)_{L} → ⟨y⟩_{R} + (A)_{L} defined by x · r + c ↦ y · r + c is well-defined and it is an isomorphism of R-modules. Since ⟨x⟩_{R} + (A)_{L} and ⟨y⟩_{R} + (A)_{L} are closed under the λ_{i} (see Proof of Proposition 3.6 in [8]), they are isomorphic L-structures over A. Therefore x and y have same type over A, which is absurd because x ∈ (A)_{def}. We can conclude that q ∈ K and thus that x ∈ (A)_{L}, a contradiction. □

5. Description of types. Finite imperfection degree.

In this section, we are going to establish a correspondence between p.p. complete L_{R}-types of T (e) and certain submodules of the free right R-module generated by countably many variables.

Notation 5.1. Let x_{∞} be the ω-tuple of variables (x_{i} : i ∈ (p^{e})^{<ω}) = (x, x_{0}, …, x_{p^{e}−1}, x_{00}, ⋯) and S any ring extension of R. We denote by ⟨x_{∞}⟩_{S} the free right S-module generated by x_{∞} and by I_{S}^{0} be the right S-submodule of ⟨x_{∞}⟩_{S} generated by all the expressions of the form x_{j} − ∑_{i∈p^{e}} x_{(j,i)} · t · m_{i}, for any indeterminate x_{j}. When the context is clear, we will allow ourselves to drop the subscript S.

Lemma 5.2. One can define unary functions λ_{i} on ⟨x_{∞}⟩_{R}/I^{0} in such a way that it becomes a model of T_{λ}. This L_{R}-structure can in turn be embedded in a model of T^{tf} which is of the form ⟨x_{∞}⟩_{R.X^{−1}}/I_{R.X^{−1}}^{0}, where X is the set of separable elements of R.

Proof: A typical element of ⟨x_{∞}⟩_{R} is of the form ∑_{i} x_{i} · r_{i}, r_{i} ∈ R. We define λ_{i}(x_{j}) as x_{ji} and we extend λ_{i} on ⟨x_{∞}⟩_{R} proceeding as in Lemma 3.2 in [8]. So, ⟨x_{∞}⟩_{R}/I^{0} becomes an L-structure. Note that it is torsion-free: suppose that t(x_{∞}) · r ∈ I^{0}, using the generators of I^{0}, we may assume that any x_{j} occurring in t(x_{∞}) has the property that j is of fixed length ℓ, then we use that ⟨x_{∞}⟩_{R} is torsion free and that I^{0} ∩ ⟨(x_{i})_{i∈(p^{e})^{ℓ}}⟩_{R} = 0. Then to embed it in a model of T^{tf}, we apply Proposition 4.3 (2).

Let (I^{0})^{e} := {n·z^{−1} : n ∈ I^{0}, z ∈ X} = I_{R.X^{−1}}^{0} be the extension of I^{0} in ⟨x_{∞}⟩_{R}·X^{−1}. Since ⟨x_{∞}⟩_{R}/I^{0} is X-torsion-free, ((I^{0})^{e})^{c}= I^{0}, where (.)^{c} denotes the contraction i.e. if J is a right R.X^{−1}-submodule of ⟨x_{∞}⟩_{R} ·X^{−1} then J^{c} := {m ∈ ⟨x_{∞}⟩_{R} : m·1^{−1} ∈ J} (see [11] Theorem 9.17). We have, by uniqueness of the R.X^{−1}-structure (see [11] Proposition 9.12), that ⟨x_{∞}⟩_{R} · X^{−1}/I_{R}^{0} · X^{−1} ≅ ⟨x_{∞}⟩_{R.X^{−1}}/I_{R.X^{−1}}^{0}. □

By the previous lemma, ⟨x_{∞}⟩_{R}/I^{0} is equipped with a ring structure, multiplication by x and elements of K being given by the R-module structure and each x_{i} acting like λ_{i}. This ring acts naturally on each model of T (e).

Proposition 5.3. ⟨x_{∞}⟩_{R}/I^{0} is the ring of definable endomorphisms of any model of T (e).

Proof: By 4.6 any definable endomorphism is of the form x ↦ τ_{x}(x) for terms τ_{x} from ⟨x_{∞}⟩_{R}. By a compactness argument τ_{x} does not depend on x. Now a term τ ∈ ⟨x_{∞}⟩_{R} vanishes identically on some (any) model of T (e) iff it belongs to I_{0} (for more details on these two last assertions, see [1] 3.3.1). □

The following definitions are made to reflect what happens in the field case (see [7] p.148-149).

Definition 5.4. Let M be an R-module. An R-submodule I of M is called separable if whenever it contains an element of the form ∑_{i∈p^{e}} y_{i} ·t·m_{i} with y_{i} ∈ M, it contains every element y_{i} for i ∈ p^{e}.

So, if M is in addition a model of T_{λ}, any separable R-submodule of M is closed under the λ-functions.

Lemma 5.5. (1) Let I be a separable R-submodule of an R-module M. If a · q belongs to I for some a ∈ M and q ∈ R − {0}, then a · q′ belongs to I for some q′ ∈ X.

(2) Assume furthermore that M is a model of T (e), then the separable closure I^{sep} of I is again separable.

(3) Let I be a non zero right ideal of R and suppose it is separable. Then it is generated by a separable polynomial.

Proof:

1. If q ∉ X, decompose it as in Lemma 3.2 in [8]: q = t.q′ = ∑_{i} \sqrt[p]{q′_{(i)}}.t.m_{i}. We

have that a · t · q′ = ∑_{i} a · \sqrt[p]{q′_{(i)}}.t.m_{i}, so by separability of I, we have that for each

i, a · \sqrt[p]{q′_{(i)}} belongs to I. In case of q′ separable, for some i, we have that \sqrt[p]{q′_{(i)}} is separable, if not we iterate the procedure.

2. Let a belong to I^{sep}, then there exists q ∈ X such that a · q ∈ I. So as in the proof of Proposition 3.6 (3) in [8], λ_{i}(a) belongs to ⟨I, a⟩_{R}.

3. The ring R is right principal and so I is generated by a polynomial q ≠ 0. Suppose that q = t.q′, then by Lemma 3.2 in [8], q = t.q′ = ∑_{i} \sqrt[p]{q′_{(i)}}t.m_{i}. So, since

I is separable, each \sqrt[p]{q′_{(i)}} belongs to I, but they are of degree smaller than q, a contradiction. □

Let M be a substructure of M and assume that it is a model of T_{λ}. We will denote by M_{L} the expansion of M to an L-substructure of M_{L}. Now we show that there is a bijection between 1-types over M and some right R-submodules of M⟨x_{∞}⟩ := ⟨x_{∞}⟩_{R} ⊕ M.

Notation 5.6. Let a ∈ M, set a_{∞} := (λ_{j}(a) : j ∈ (p^{e})^{<ω}), let p(a, M) be the set of all L_{R}-formulas satisfied by a in M over M; it is axiomatized by the p.p. complete L_{R}- type p_{+}(a, M) of a over M, or equivalently-using the positive quantifier elimination for T_{e} (see section 2)- by the set p_{+}(a, M_{L}) of all atomic L-formulas satisfied by a over M in the expansion M_{L}. We will denote by S_{1}(M) the set of complete 1-L_{R} types with parameters in M.

To the L-type p_{+}(a, M_{L}), we associate:

- the (a priori partial) L_{R}-type p_{+}(a_{∞}, M) in countably many variables, obtained by substituting x_{j} for λ_{j}(x) in p_{+}(a, M_{L}),

- the R-submodule of M⟨x_{∞}⟩

I(a_{∞}, M) := {t(x_{∞}) + m ∈ M⟨x_{∞}⟩; M ⊧ t(a_{∞}) + m = 0}

(observe that, if e > 0, independently of M, I(a_{∞}, M) has the cardinality of R),

- and the R-submodule of R⟨x_{∞}⟩,

I_{R}(a_{∞}, M) := {t(x_{∞}) ∈ ⟨x_{∞}⟩_{R}; t(a_{∞}) ∈ M}.

For a ∈ M and q := tp(a, M), we write I(q) and I_{R}(q) for I(a_{∞}, M) and I_{R}(a_{∞}, M).

Lemma 5.7. Let a, b ∈ M, the following are equivalent:

(1) tp(a, M) =tp(b, M)

(2) p_{+}(a, M_{L})=p_{+}(b, M_{L})

(3) I(a_{∞}, M) = I(b_{∞}, M)

Proof: (1)⇒(3) is clear. (3)⇒(2) Observe that, by Corollary 3.4 in [8], any atomic formula in p_{+}(a, M_{L}) is equivalent to one of the following form ∑_{i∈p^{en}} λ_{i}(x) · d_{i} + m = 0, for some n ≥ 1, d_{i} ∈ R and m ∈ M. (2)⇒(1) by the positive quantifier elimination. □

Proposition 5.8. (1) If M and N be models of T (e), M ⪯ N ⪯ M, and a ∈ M. Let t(a, N) ∈ S_{1}(M). Then, t(a, N) does not fork over M iff I(a_{∞}, M) = I(a_{∞}, N) iff I_{R}(a_{∞}, M) = I_{R}(a_{∞}, N).

(2) If M and N are L-substructures of M_{L}, M ⊆ N, and a ∈ M, then t(a, N) does not fork over M iff any element in I(a_{∞}, N) has a multiple in I(a_{∞}, M) iff any element in I_{R}(a_{∞}, N) has a multiple in I_{R}(a_{∞}, M).

For a definition of forking, see [2], page 3 and [18] p.111.

Proof: 1. First, I(a_{∞}, M) = I(a_{∞}, N) and I_{R}(a_{∞}, M) = I_{R}(a_{∞}, N) are trivially equivalent when M ⊆ N. Now, if t(a, N) does not fork over M then t(a, M) should not represent more formulas than t(a, N), hence I_{R}(a_{∞}, M) = I_{R}(a_{∞}, N). The converse follows by 5.7 and uniqueness of non forking extensions over models.

2. Follows from 1 and the following facts and their analogues for the I_{R}’s.

- If M and N are models, ”any element in I(a_{∞}, N) has a multiple in I(a_{∞}, M)” is equivalent to ”I(a_{∞}, M) = I(a_{∞}, N)”.

- For N = ⟨⟨M⟩⟩, since M is an L-structure, N = M^{sep} and any element in I(a_{∞}, N) has a multiple in I(a_{∞}, M).

- Since the prime model over some set of parameters is algebraic over these parameters, t(a, N) does not fork on M iff t(a, ⟨⟨N⟩⟩) does not fork on ⟨⟨M⟩⟩. □

Definition 5.9. (1) We call standard an R-submodule of M⟨x_{∞}⟩ which includes I^{0} (see Notation 5.1).

(2) Now, assume that M contains K^{sep}. Then, an R-submodule I of M⟨x_{∞}⟩ is called separably closed if whenever v · q ∈ I, for some q ∈ X and v ∈ M⟨x_{∞}⟩, then v − c ∈ I, for some c ∈ K^{sep}.

Now we will further assume that M contains K^{sep}, in other words all the torsion elements. We will characterize submodules of the form I(a_{∞}, M) amongst submodules of M⟨x_{∞}⟩.

Proposition 5.10. Let M be an R-submodule of M. Suppose that M is a model of T_{λ} containing K^{sep}, and let I ⊆ M⟨x_{∞}⟩ be a right R-submodule. Then there is a ∈ M such that I = I(a_{∞}, M) iff I is standard, separable, separably closed and such that I ∩ M = {0}.

Proof: That I(a_{∞}, M) ∩ M = {0} is formally clear. Now, I(a_{∞}, M) is standard because M is a model of axiom 4, separable because M is a model of axiom 5, and separably closed because M is a model of axiom scheme 3.

Conversely let I be as stated. Let M^{sep} be the separable closure of M in M, S := R.X^{−1} be the ring of fractions of R with respect to the set X of separable elements of R and M̃ be the module M^{sep} ⊕ ⟨x_{∞}⟩_{S}/I_{S}^{0}. It is a model of T (e) since M^{sep} is a model of T (e) and ⟨x_{∞}⟩_{S}/I_{S}^{0} is a model of T^{tf} (see the proof of Lemma 5.2). Consider the quotient N = M̃/(I/I_{S}^{0}). As M ∩ I = {0}, M embeds in N. Since I is standard and separable, N is a model of T_{λ} (and so the torsion is equal to the separable torsion). Since I is separably closed, N_{tor} is isomorphic to K^{sep} and can be identified with the copy of K^{sep} which is in M. Since M̃ satisfies axiom scheme 2, a fortiori N satisfies it. So N is a model of T (e) containing M and which contains an element x + I with the property that I((x + I)_{∞}, M) = I. The result follows by |R|^{+}-saturation of M. □

Corollary 5.11. Let M and a be as above. The map a ↦ I(a_{∞}, M) defines a bijection between 1-types over M and right R-submodules I of M⟨x_{∞}⟩ standard, separable and separably closed with I ∩ M = {0}. □

Remark. Note that, for M as above and q ∈ S_{1}(M), the proof of 5.10 describes the prime model M⟨⟨q⟩⟩ over M and q.

Corollary 5.12. Let M and N be models of T (e), M ⪯ N ⪯ M, and a ∈ M. Then, t(a, N) does not fork over M iff N⟨⟨a⟩⟩/N = M⟨⟨a⟩⟩/M for the canonical inclusion.

Proof: Clear from 5.8 and proof of 5.10. □

We describe now types over R-submodules M of models of T (e), with M a model of T_{λ}, without assuming that M contains K^{sep}.

Definition 5.13. (1) A R-submodule I of M⟨x_{∞}⟩ is called compatible if, for each q ∈ X, there are at most p^{deg(q)} elements t(x_{∞})+m ∈ M⟨x_{∞}⟩ different modulo I, such that (t(x_{∞})+m)·q ∈ I. If K^{sep} ⊆ M and I ∩M = {0}, it is equivalent to be separably closed.

(2) An R-submodule I of ⟨x_{∞}⟩_{R} is called X-saturated if whenever v · q ∈ I, for some q ∈ X and v ∈ ⟨x_{∞}⟩_{R}, then v ∈ I (see p. 62 in [19]). Note that if in addition I is separable, then the properties of being X-saturated and (R − {0})-saturated are equivalent (see Lemma 5.5 (1)).

Proposition 5.14. Let M be a L-substructure of M_{L} and a ∈ M. The map a ↦ I(a_{∞}, M) defines a bijection between 1-types over M and right R-submodules I of M⟨x_{∞}⟩ which are standard, separable, compatible and satisfy I ∩ M = {0}.

Proof: It is clear that a module of the form I(a_{∞}, M) has the properties above. Conversely let I be such. Claim: An R-module N ⊇ M and of small torsion i.e. containing, for each separable q ∈ R, at most p^{deg(q)} elements x satisfying x · q = 0, embeds over M in a model of T_{sep}, which can be taken minimal over N. Proof of the Claim: After the techniques from [8] 5.3, 5.4 and 5.8, we decompose N = ∪_{i∈ω+1}N_{i} where N_{0} = M, the union is increasing and continuous, and each N_{i} contains all elements m ∈ N satisfying m · q = 0 for some separable q ∈ R of degree < i, and we embed inductively each N_{i} in any |N|^{+}-saturated model of T_{sep}. By [8] 5.8, there is a minimal model N^{sep} ⊧ T_{sep} containing N, which can be taken minimal over N.

Now, N := M⟨x_{∞}⟩/I has small torsion and is provided with an L-structure extending M. By the claim it embeds in N^{sep} ⊧ T_{sep}, by [8] 5.9 N^{sep} has a unique L-structure extending N, and, as such, N^{sep} ⊧ T_{e}. Clearly I(x_{∞} + I, M) = I. □

Proposition 5.15. Let M be a model of T (e). Then, a submodule I of ⟨x_{∞}⟩_{R} is of the form I_{R}(a_{∞}, M) for some a ∈ M iff it is is standard, separable and X-saturated iff it is of the form I_{R}(a_{∞}, N) for some a ∈ M and N_{L} ⪯ M_{L}.

Proof: The second equivalence follows from the first one, which we get by the previous proposition, with M = 0. □

Notation 5.16. By Lemma 5.7, the submodule I(a_{∞}, M) (respectively I_{R}(a_{∞}, M)) of M⟨x_{∞}⟩ (respectively ⟨x_{∞}⟩_{R}), a ∈ M, only depends on the type q ∈ S_{1}(M) of a over M. We will denote it by I(q) (respectively I_{R}(q)). By Proposition 5.14 (respectively 5.15), any submodule I of M⟨x_{∞}⟩ (respectively ⟨x_{∞}⟩_{R}) which is standard, separable, compatible (respectively X-saturated) and satisfies I ∩ M = {0} is of that form.

We will call type submodule any such submodule of ⟨x_{∞}⟩_{R} of the form I_{R}(q).

If we consider the theory T^{tf} instead of T (e), using analogous proofs to the ones in this section, we can also obtain a correspondence between 1-types over a model M and standard, separable and X-saturated submodules I of M⟨x_{∞}⟩ satisfying I ∩M = {0}.

6. Types and infinitely definable subgroups

Classically, one associates with a type q with parameters over a subset A, an infinitely definable group without parameters and its connected component (see [18], chapter 2, p. 25-26). Whenever, q is a type over a model, those two infinitely definable groups coincide (see [18] lemma 2.6) and in that case the free part of q (see [18] p.134, 135) is the generic type of the associated subgroup.

We will first show that the connected component of a definable subgroup of (M, +, 0), where M ⊧ T (e) is still definable.

Then, we will prove an analog of the Nullstellensatz in our context using on one hand the description of the type submodules given in the previous section and on the other hand the generic types.

Finally, thanks to the fact that we do have prime models, we will show a lattice isomorphism between type submodules and models.

All the groups G we will be considering are subgroups of (M, +, 0), where M is a model of T (e). To stress that fact we will use the phrase additive group, also we will say that a group is ∅-definable when it is definable without parameters. Let us first recall some terminology.

A subgroup H is relatively definable in a group G if it is the intersection of a definable subset with G.

An infinitely definable group is connected iff it does not contain any relatively definable proper subgroup of finite index. The connected component, usually denoted by G^{0}, of an infinitely definable group G is the intersection of all relatively definable subgroups of finite index. In a stable theory T, it is infinitely definable using the same parameters as the ones used for defining G and has index at most 2^{|T |} ([20] p.61), it is the intersection of all relatively definable subgroups of bounded index. We prove in the following proposition that we have more in our context: in case G is definable, G^{0} has finite index in G i.e. G is connected-by-finite. This implies that, for a definable G, G^{0} itself is definable, and not only infinitely definable.

Proposition 6.1. Any additive subgroup definable in T (e) is connected-by-finite.

Proof: We can deduce this from the corresponding result for separably closed fields (see Corollary 4.6 in [3]), but we give here a direct proof.

First, we note that in any theory of modules, to show that a definable group is connected-by-finite, it suffices to prove it for p.p. definable groups. Indeed, one observes using B.H. Neumann’s Lemma ([13] p. 140) that any definable group has a p.p. definable subgroup of finite index.

Using positive elimination for T_{e} in L (6.9 and remark 6 in [8]), the fact that, for each integer n and M_{L} ⊧ T_{e}, (λ_{i})_{i∈p^{en}} defines an isomorphism between (M, +) and (M^{p^{en}}, +), and the triangularization of matrices (6.4 in [8]), any subgroup of some Cartesian power of (M, +) is definably isomorphic to a group G defined by a system x̄ · A = ¯0, where A is a lower triangular (m × k)-matrix with only separable, hence non zero, elements on its diagonal. Let r_{0}, ⋯, r_{k−1} be these coefficients. Let us prove now, by induction on k independently of m, that G is connected-by-finite, with [G : G^{0}] ≤ ∑_{i∈k} p^{deg(r_{i})}. It is clear for k = 0 as, in this case, G is the additive group M^{m}. For k ≥ 1, consider the definable additive homomorphism f : (x_{i})_{i∈m} ↦ (x_{0} · r_{0}, x_{1}, ⋯, x_{m−1}). The image of G under f lies in the group H := {x̄ ∈ M^{m}; x̄·B = ¯0}, where B is the matrix obtained from A by replacing r_{0} by 1. Because r_{0} is separable and M_{L} models T_{e}, f is surjective on H and its kernel is finite, of cardinality p^{deg(r_{0})}. Now H is isomorphic to M × G′ where G′ := {x̄ ∈ M^{m−1}; x̄ · A′ = ¯0}, for A′ the (m−1)×(k−1)-matrix obtained from A by removing the first line and the first column.

By induction ([H : H^{0}] =)[G′ : G′^{0}] ≤ ∑_{i=1}^{k−1} p^{deg(r_{i})} and G is finite-by-(connected-by finite). As it is stable, it implies that it is connected-by-finite (we can also prove it directly). Indeed, f(G^{0}) is a connected subgroup of H of bounded index, hence f(G^{0}) = H^{0}, which also proves [G : G^{0}] ≤ [H : H^{0}] + | ker f| ≤ ∑_{i∈k} p^{deg(r_{i})}. □

Note that a module is connected iff it does not contain any proper p.p. definable subgroup of finite index (see Lemma 1, paragraph 6, part 2, p. 155 in [12].)

In an arbitrary module, a connected definable subgroup is p.p. definable (apply Neumann’s Lemma) and so ∅-definable; here, we will prove that any additive subgroup ∅-definable in T (e) is p.p. definable. However, it is not known whether this holds in general (see [18], exercise 2 on page 118).

Proposition 6.2. Any additive subgroup ∅-definable in T (e) is p.p. definable.

Proof: Let G be an additive definable subgroup. We argue in the prime model K^{sep}. As we just have seen G is connected-by-finite, hence (p.p. definable)-by-finite. Since all groups are here vector spaces over 𝔽_{p}, G is the product of G^{0} by a finite group F, which is a finite additive subgroup of K^{sep}. Consider the group F′ = Σ{σF; σ ∈ Gal(K^{sep}; K)}. Since G is ∅-definable without parameters, G = G^{0} + F′. Now F′, as a finite additive subgroup of K^{sep}, is the zero set of some additive polynomial over K^{sep}, and in fact over K as it is invariant under Gal(K^{sep}; K), hence the r-torsion for some r ∈ R. □

After these preliminaries, we relate the description of types in T (e) we have given in section 5 to the classical description of types in modules.

Definition 6.3. We define (see [18] pages 24, 25 and 26).

G := {definable additive subgroups of M} G_{0} := {connected definable additive subgroups of M} Ḡ_{0} := {connected infinitely definable additive subgroups of M} T := {types submodules of ⟨x_{∞}⟩_{R}},

and for M ⪯ M and q ∈ S_{1}(M),

G(q) := {G ∈ G; q ⊢ x − a ∈ G for some a ∈ M}
G_{0}(q) := {G ∈ G_{0}; q ⊢ x − a ∈ G for some a ∈ M}
T (q) := {N ∈ T ; I_{R}(q) ⊆ N}
G(q) := ∩{G; G ∈ G(q)}.

Equipped with the operations connected component of the intersection, and of the sum, Ḡ_{0} is a complete lattice. It happens here that G_{0} is a sublattice. Indeed, the intersection and the sum of definable groups are definable, and their connected components too, by 6.1. Ḡ_{0} is in fact the completion of G_{0}.

As a consequence, because we are considering types over a model M where all classes of G/G^{0} exist, we have that G ∈ G(q) iff G^{0} ∈ G_{0}(q), and G(q) is also the intersection of all groups in G_{0}(q). Note that, if tp(x, M) is the generic of some group G ∈ Ḡ_{0}, then G = G(x, M) (we use here the notation G(x, M) for G(tp(x, M))). Indeed G(x, M) ≤ G by definition of G(x, M), and q is stable under (over M) independent addition because it is a principal generic ([17] p. 163), hence G(q) = {G ∈ G; q ⊢ x ∈ G}; now x is in no proper definable subgroup of G. Hence Ḡ_{0} = {G(q); q a type over some model of T (e)}, (see also [18], where Ḡ_{0} appears as PP_{0}^{(1)}, in particular Lemma 5.10). Our description of types is related to the classical one in the following way: for q ∈ S_{1}(M) and t ∈ ⟨x_{∞}⟩_{R}, t ∈ I_{R}(q) iff t(a_{∞}) ∈ M for any a realizing q, iff {y; t(y_{∞}) = 0} ∈ G(q).

Proposition 6.4. Suppose M is a |K|^{+}-saturated model of T (e). Then the maps

Ḡ_{0} → T : G → I_{R}(G) := {t ∈ ⟨x_{∞}⟩_{R} ; ∀g ∈ G t(g_{∞}) = 0}
T → Ḡ_{0} : N → V (N) := {x ∈ M ; ∀t ∈ N t(x_{∞}) = 0}

are inverse bijections between Ḡ_{0} and T . Their restrictions are inverse bijections between {G ∈ Ḡ_{0}; G ≤ G(q)} and T (q). For G ∈ Ḡ_{0}, the generic type of G is the unique type q over M satisfying I_{R}(G) = I(q).

Proof: For N ∈ T , H := ⟨x_{∞}⟩_{R}/N is a torsion-free L-structure which embeds in any |K|^{+}-saturated model of T_{e}, is as such a subset of V (N) and satisfies I_{R}(H) = N. Hence I_{R}(V (N)) = N, this is the analogue in the module framework of the Nullstellensatz for separably closed fields. It is clear that V (N) is an infinitely definable subgroup of M, let us prove now it is connected. Since N is X-saturated the condition N = I(q) defines a unique type q ∈ S_{1}(M). As we already noticed, for t ∈ ⟨x_{∞}⟩_{R}, t ∈ I_{R}(q) iff {y; t(y_{∞}) = 0} ∈ G(q). Hence G(q) = V (N), but we know that G(q) is connected. Now q is clearly contained in V (N) and stable under addition of any element of M ∩ V (N), hence it is the generic of this group.

Conversely, by quantifier elimination in the language L, any G ∈ G_{0} is the zero set of terms of ⟨x_{∞}⟩_{R} and I_{R}(G) is clearly standard and separable. We show now it is X-saturated if G is connected. Let t(G) · g = 0 for some t ∈ ⟨x_{∞}⟩_{R} and a separable polynomial g (of degree d). Then the condition ”t(y_{∞}) = 0 ∧ y ∈ G” defines a subgroup of finite index (≤ d) of G, hence by connexity G itself, hence t ∈ I_{R}(G). □

For q ∈ S_{1}(M), let q_{G} the generic type over M of G(q) (as q is a type over a model, q_{G} is the “free part” q_{∗} of q in [18] section 6.1), hence I_{R}(q) = I(q_{G}) and G(q) = G(q_{G}). The construction of the minimal model realizing some type allows us to understand an interesting connection between q and q_{G}.

Proposition 6.5. Let M be a model of T (e). Then, the models M⟨⟨q_{G}⟩⟩ and M⟨⟨q⟩⟩ are isomorphic over M iff M is a direct summand in M⟨⟨q⟩⟩. This is in particular the case when in addition M is pure-injective.

Proof: By the construction of M⟨⟨q⟩⟩ in 5.10, if I(q) = I_{R}(q) then M⟨⟨q⟩⟩ is equal to M ⊕ ⟨x_{∞}⟩_{R.X^{−1}}/I_{R}(q).X^{−1} and so M is a direct summand in M⟨⟨q⟩⟩.

In particular, note that M is always a direct summand in M⟨⟨q_{G}⟩⟩.

In the other direction, M⟨⟨q_{G}⟩⟩/M = M⟨⟨q⟩⟩/M and M is a direct summand in M⟨⟨q⟩⟩ exactly when M⟨⟨q_{G}⟩⟩ and M⟨⟨q⟩⟩ are M-isomorphic over M. □

Given two models M ⪯ N ⪯ M such that M is direct summand in N, then the set of models P, M ⪯ P ⪯ N, equipped with sum and intersection, is a complete lattice, as is T (q).

Proposition 6.6. The lattice of models between M and M⟨⟨q_{G}⟩⟩ and T (q) are isomorphic.

Proof: Define

S^{tf}(q) := {N; N is a model of T^{tf}, N ⪯ ⟨x_{∞}⟩_{S}/(I_{R}(q) · X^{−1})}
S(q_{G}) := {N; N is a model of T (e), M ⪯ N ⪯ M⟨⟨q_{G}⟩⟩},

and consider the maps α : N ∈ T (q) ↦ (N · X^{−1})/(I_{R}(q) · X^{−1}) ∈ S^{tf}(q) β : N ∈ S^{tf}(q) ↦ M ⊕ N ∈ S(q_{G}) γ : N ∈ S(q_{G}) ↦ (N/M) ∩ (⟨x_{∞}⟩_{R}/I_{R}(q)),

this intersection being taken in ⟨x_{∞}⟩_{S}/I_{R}(q) · X^{−1}, where N/M canonically embeds. Since modules in T (q) are X-saturated, one has γ ∘ β ∘ α = Id. □

7. Dependence relations

In this section, we are going to consider several notions of dependence and investigate the relationships among them. The first ones are algebraically defined and use either the fact that the ring is right Ore or that X is a right denominator set. Then we will consider relations which are usually defined in the context of modules using the p.p. elimination. In general, those are not transitive (nor reflexive) unless if we restrict to particular subsets. And finally the relation of forking which is again in general non transitive. Using our prime model construction, we will be able to obtain tighter connections than usual.

We call here dependence relation over some abstract structure N a relation of the form b ≺ A, for b ∈ N and A ⊆ N, satisfying the following axioms:

- If b ∈ A, then b ≺ A (reflexivity)

- b ≺ A iff b ≺ A_{0} for some finite A_{0} ⊆ A (finite character)

- if b ≺ A ∪ {c} and c ≺ A then b ≺ A (transitivity)

- if b ≺ A ∪ {c} and b ⊀ A then c ≺ A ∪ {b} (symmetry, or exchange property). These axioms allow us to extend coherently the dependence relation to subsets:

{b_{1}, …, b_{k}} ≺ A iff, for some i = 1, …, k − 1, b_{i+1} ≺ A ∪ {b_{1}, …, b_{i}} and,
B ≺ A iff for any finite B_{0} ⊆ B, B_{0} ≺ A,

and to localize to a set of parameters: x ≺_{A} B: iff x ≺ B ∪ A. The corresponding closure is defined as follows: cl(A) = {x ∈ N; x ≺ A}, and it characterizes the relation ≺.

To a dependence relation is naturally associated a notion of dimension of a set as the cardinality of a basis, i.e. a maximally independent subset. This is well defined since all bases have the same cardinality ([14] p.122).

N is now a model of T (e) or T^{tf} and let x ∈ N.

Definition 7.1. (1) x is R-dependent on A iff there exist a_{1}, ⋯, a_{n} in A and non

zero elements r_{0}, r_{1}, ⋯, r_{n} in R, such that x · r_{0} = ∑_{i≥1} a_{i} · r_{i}.

(2) x is separably R-dependent on A, x ≺ A, iff there exist a_{1}, ⋯, a_{n} in A and separable elements r_{0}, r_{1}, ⋯, r_{n} in R, such that x · r_{0} = ∑_{i≥1} a_{i} · r_{i}.

Proposition 7.2. (1) The R-dependence is a dependence relation.

(2) The separable R-dependence is a dependence relation. The corresponding closure of A is ⟨A⟩_{R}^{sep}.

(3) If A is an L-substructure, then R-dependence and separable R-dependence on A coincide.

Proof: Reflexivity, finite character and symmetry are clear for both relations. Transitivity of the first one follows from the right Ore property of R: two non zero elements have a non zero common multiple. Transitivity of the second one follows from the fact that X is a right denominator set of R: two separable elements in R have a separable common multiple. The description of both closures is clear, and also the fact that the separable dependence is stronger. The converse in case A is closed under the functions λ follows from Lemma 3.2 item 3 in [8]. □

Amongst relations arising from stability theory, the relation “t(x, A) forks over ∅” (“x ↓̸ A”) is in general not transitive, and the relation “x is model theoretically algebraic over A” not symmetric. But they are when restricted to minimal types, in which case they coincide and give rise to a dependence relation. This is a basic fact in stability. Existence and description of prime models give us a bit more in our case (recall that (A)_{L} denotes the L-substructure generated by A). In fact one has to be careful since, for an algebraic x, one has x ↓ x, which means that ↓̸ is not reflexive. Classically this fact is handled by a quotient identifying x to its algebraic closure. We proceed here slightly differently and consider the relation “x is algebraic or x ↓̸ A”.

Proposition 7.3. Let x ∈ M and A, B ⊆ M be such that RU(x, A) and RU(b, A) are at most 1, for each b ∈ B. Then the relation “RU(x, A) = 0 or x ↓̸_{A} B” is a dependence relation, the corresponding closure of B is the model theoretical algebraic closure of A ∪ B, which is also the prime model ⟨⟨A ∪ B⟩⟩ over A ∪ B.

Without restricting ourselves to minimal types there are some classical algebraic characterizations of forking, which we can in our case make more precise or more general. For example let us consider the independence relation introduced by Garavaglia, which works over any ring and which we will here call the G-independence (see for instance [22], p.177).

Definition 7.4. Two subsets A, B of a pure injective R-module N are G-independent if for any p.p. formula ϕ(x̄, ȳ) such that ϕ(ā, b̄) holds in N with ā ∈ A, b̄ ∈ B, then ϕ(¯0, b̄) holds, or, equivalently, if there is a decomposition N = N_{1} ⊕ N_{2}, with A ⊆ N_{1} and B ⊆ N_{2}.

Note that if two subsets of parameters A and B included in a pure-injective model N are G-independent, then they remain G-independent in any pure-injective extension of N in which N is pure. So, in the case when N is a model of T (e) or T^{tf}, this property is invariant under taking L-extensions.

Proposition 7.5. In T^{tf}, we have the following equivalence: A ↓_{∅} B iff A and B are G-independent.

Proof: Forking and G-independence are linked in any theory of modules closed under direct sums, by the above equivalence (see Corollary 11.2 in [22]). □

Lemma 7.6. Let A, B be included in M_{L}. Then, (A)_{L} ∩ (B)_{L} = {0} (i.e. (A)_{L} and (B)_{L} are R-independent) iff A and B are G-independent.

Proof: (→) Let ϕ(x̄, ȳ) be a p.p. formula such that ϕ(ā, b̄) holds with ā ∈ A, b̄ ∈ B. Since T_{e} admits positive quantifier elimination ϕ(x̄, ȳ) is equivalent to a conjunction of L-terms t_{i}(x̄, ȳ) = 0. By Lemma 3.3 and Corollary 3.4 in [8], we have that t_{i}(x̄, ȳ) = t_{i1}((x̄_{j})_{j∈p^{en}}) + t_{i2}((ȳ_{j})_{j∈p^{en}}) for some integer n and t_{i1}, t_{i2} two L_{R}-terms. By hypothesis, we get that t_{i1}((ā_{j})_{j∈p^{en}}) = 0 and since t_{i2}((¯0_{j})_{j∈p^{en}}) = 0, we have that ϕ(ā, ¯0) holds. (←) Conversely, suppose that A and B are G-independent and suppose that a = b, with a ∈ (A)_{L} and b ∈ (B)_{L}. Since a and b are L-terms in respectively A and B, we get that some p.p. formula in L_{R} holds for ā and b̄ in respectively A and B, then it should hold for ā and ¯0, hence a = 0. □

Lemma 7.7. Let A, B and C be included in M_{L}. Then, ((B)_{L} + ⟨⟨A⟩⟩) ∩ ((C)_{L} + ⟨⟨A⟩⟩) = ⟨⟨A⟩⟩ iff B + ⟨⟨A⟩⟩ and C + ⟨⟨A⟩⟩ are G-independent in T_{free}^{tf}.

Proof: The proof is the same for (B)_{L} + ⟨⟨A⟩⟩ and (C)_{L} + ⟨⟨A⟩⟩ in T_{free}^{tf} as for the previous proposition, since T_{free}^{tf} admits also positive quantifier elimination in L. □

Proposition 7.8. In T (e), B ↓_{A} C iff (B)_{L} and (C)_{L} are independent in the sense of ≺_{A} iff (B)_{L} + ⟨⟨A⟩⟩ ∩ (C)_{L} + ⟨⟨A⟩⟩ = ⟨⟨A⟩⟩ iff (B)_{L} + ⟨⟨A⟩⟩ and (C)_{L} + ⟨⟨A⟩⟩ are G-independent in M_{L}/⟨⟨A⟩⟩.

Proof: The first equivalence is trivial since any point of (B)_{L} is definable over B, the last one is the previous lemma, let us prove the middle one. We use the following criterion, true in any complete theory of modules (see [18] p.113): b̄ ↓ā c̄ iff for any p.p. formula ψ such that M ⊧ ψ(b̄, c̄, ā) there exists a p.p. formula ϕ and a tuple of elements ā′ ⊆ ā such that M ⊧ ϕ(b̄, ā′) and the index [ϕ(v̄, ¯0) : ϕ(v̄, ¯0) ∧ ψ(v̄, ¯0, ¯0)] is finite.

Set M_{0} := M_{L}/⟨⟨ā⟩⟩, note that it is now a model of T_{free}^{tf}. Let us denote the class in M_{0} of any element d ∈ M by d_{≡} := d + ⟨⟨ā⟩⟩. (⟶) Suppose that M_{0} ⊧ χ(b̄_{≡}, c̄_{≡}), where χ is a p.p. formula. By positive quantifier elimination in T_{free}^{tf}, there exist L-terms t_{i}, s_{i} such that χ(ū, v̄) ⟷ ⋀_{i} t_{i}(ū)+s_{i}(v̄) = 0. So, there exist L-terms r_{i} and separable elements q_{i} ∈ X such that M_{L} ⊧ ⋀_{i}(t_{i}(b̄)+ s_{i}(c̄)) · q_{i} = r_{i}(ā). The L-formula ⋀_{i}(t_{i}(v̄) + s_{i}(w̄)) · q_{i} = r_{i}(ū) is equivalent mod T_{λ} to a p.p. L_{R}-formula ψ(v̄, w̄, ū). By the criterion, there exists a p.p. formula ϕ and ā′ ⊆ ā such that M ⊧ ϕ(b̄, ā′) and the index [ϕ(v̄, ¯0) : ϕ(v̄, ¯0)∧ψ(v̄, ¯0, ¯0)] is finite. So, M_{0} ⊧ ϕ(b̄_{≡}, ¯0_{≡}). But in M_{0}, if the index of two p.p. definable subgroups is finite, it is equal to 1. So, M_{0} ⊧ ψ(b̄_{≡}, ¯0_{≡}, ¯0_{≡}). So, M_{0} ⊧ ⋀_{i}(t_{i}(b̄) + s_{i}(¯0)) · q_{i} = r_{i}(¯0) = 0. But M_{0} is torsion-free and so M_{0} ⊧ χ(b̄_{≡}, ¯0_{≡}). (⟵) Let ψ be a p.p. L_{R}-formula such that M ⊧ ψ(b̄, c̄, ā). By positive quantifier elimination result in T_{e}, there exist L-terms t_{i}, s_{i}, r_{i} such that ψ(v̄, ū, w̄) is equivalent to the L-formula ⋀_{i} t_{i}(v̄)+s_{i}(ū)+r_{i}(w̄) = 0. In M_{0}, we get M_{0} ⊧ ⋀_{i} t_{i}(b̄_{≡})+s_{i}(c̄_{≡}) = 0_{≡}, which entails that M_{0} ⊧ ⋀_{i} t_{i}(b̄_{≡})+s_{i}(¯0_{≡}) = 0_{≡} and so there exist q_{i} ∈ X and Lterms r′_{i} such that M_{L} ⊧ ⋀_{i}(t_{i}(b̄) + s_{i}(¯0)).q_{i} = r′_{i}(ā). This last formula is equivalent to a p.p. L_{R}-formula and the index [⋀_{i}(t_{i}(x̄)+s_{i}(¯0))·q_{i} = r′_{i}(¯0) = 0 : ⋀_{i} t_{i}(x̄)+s_{i}(¯0) = r_{i}(¯0) = 0] is finite. □

Now using the above proposition and a result of Ziegler ([22], section 6, Lemma 6.2), we may state the weak transitivity property of ↓: Namely, let D ⊆ M, b ∈ M, let a_{1}, ⋯, a_{n+1} ∈ M and assume that in the model M/⟨⟨D⟩⟩ of T^{tf}, H(a_{i})/⟨⟨D⟩⟩, 1 ≤ i ≤ n+1, are indecomposable and that {a_{1}, ⋯, a_{n}} is ↓-independent. Then b ↓̸_{D} {a_{1}, ⋯, a_{n+1}} and a_{n+1} ↓̸_{D} {a_{1}, ⋯, a_{n}}, implies that b ↓̸_{D} {a_{1}, ⋯, a_{n}}.

8. Ranks

Let M be a |K|^{+}-saturated model of T (e) and q ∈ S_{1}(M).

Theorem 8.1. The foundation rank of G(q) in Ḡ_{0}, the foundation rank of T (q) and of the lattice of models between M and M⟨⟨q_{G}⟩⟩ are all equal to RU(q).

Proof: The interpretation of RU(q) as the foundation rank of G(q) is true in any module, see Theorem 5.12 in [18], the other characterizations follow from Propositions 6.4 and 6.6. □

We note here that we understand the modules we are considering much better than the separably closed fields. Indeed, we know that, in SCF_{e}, for a type q over an ω_{1}- saturated model M, if RU(q) is an integer n, then there is a chain of models of length n between M and M⟨⟨q⟩⟩, but we are unable to say something about the converse, or to turn the length of chain of models into a rank. Also the relation between U-rank and the depth of type ideals, the analogue in SCF_{e} of the foundation rank of T (q) for T_{e}, is unknown.

On the other hand, for M ⊧ T (e) and q a type over M, we can define, if R-dim (R-dim_{M}) denotes the dimension relative to R-dependence (with parameters from M),

RR(q) := R- dim_{M} M⟨⟨q⟩⟩ = R- dim(M⟨⟨q⟩⟩/M),

ranging in ω ∪ {∞}. This is also the maximal co-rank of a lower triangular separable matrix A such that

q(x) ⊢ x = Σ_{i∈p^{en}}x_{i} · t^{n} · m_{i} ∧ (x_{i})_{i∈p^{en}} · A = m

for some m in some Cartesian power of M. Then it is a stability rank (see 5.12), the right analogue of transcendence rank for SCF_{e}, and we get immediately from the equivalent result in SCF_{e}, the following proposition.

Proposition 8.2. There are in T_{e} minimal types of arbitrary RR-rank in ω ∪ {∞}.

We already mentioned that many results can be directly transferred from the theory SCF_{e} of separably closed fields of characteristic p and degree of imperfection e. Indeed we have the following.

Fact 8.3 ([1]). For K a definably closed subfied of C ⊧ SCF_{e}, any additive subgroup of C which is (infinitely) definable over K is also (infinitely) definable in the L_{R}- reduct of C .

Thomas Blossier also constructed in SCF_{e} (e ≥ 1) minimal additive subgroups of arbitrary transcendence degree n ∈ ω ∪ {∞}. In T (e) these groups are still infinitely definable and their U-rank can only decrease, hence it is still 1. Their generics, or more exactly their trace on L_{R} have clearly RR-rank n. Other constructions of Thomas Blossier are of interest for us, as he produced several examples of 1-based groups. On such groups the structure induced by the field is exactly the L_{R}-structure. As an example one can immediately deduce from his work that there are in T (e) 1types of any U-rank < ω^{ω}.

Let us finally remark that, as in separably closed fields, and for the same reasons, there are in T (e) no non algebraic type ranked by the Morley rank. (When K is countable, this also can be deduced from the description of the Ziegler spectrum. Indeed, let M ⊧ T^{tf} and suppose that M is ℵ_{0}-saturated. Let 𝕌_{M} be the corresponding closed set in the Ziegler spectrum. By [22] 8.6 (and see also [18] 10.19), we have that the Cantor-Bendixon rank of an element of 𝕌_{M} is equal to its m-dimension (in 𝕌_{M}) which in turn is equal to the minimum of the m-dimension of the interval of p.p. formulas [ϕ ψ] with ϕ(M) > ψ(M). By Lemma 3.4, this dimension is undefined. Now the Morley rank of q ∈ S_{1}(∅) with H(q) = N is equal to the Cantor-Bendixon rank in S_{1}(M) of a non-forking extension q̃ of q over M (see [18] p. 120)). Therefore the Morley rank of q is greater than or equal to the Cantor-Bendixon rank of H(q̃) in 𝕌_{M}. But this Cantor-Bendixon rank is undefined.

9. Elimination of imaginaries

If M is an infinite module, its theory does not admit elimination of imaginaries (e.i.), hence weaker notions have been considered (see [15] and [9]), and we will show here that T_{e} has uniform weak e.i. and uniform p.p. elimination of imaginaries (see definitions below) As a matter of comparison let us recall that, for theories which expand a theory of fields, weak e.i. implies e.i., that in the case of separably closed fields of finite imperfection degree, in order to get e.i., one needs to choose a p-basis and to add in the language constants for its elements, and that, in the infinite imperfection degree case, an appropriate language is still unknown. Some of our results will also hold for the infinite imperfection degree case (recall that the corresponding theory is denoted by T_{∞}).

Definition 9.1. (See [9], part 2). Let T be a first-order theory, ϕ(x̄, \overline{a}) be a formula and A be the set defined by ϕ(x̄, \overline{a}). Then,

(1) c̄ is a canonical base for ϕ(x̄, \overline{a}) if for all automorphism σ of the monster model of T, σ fixes A setwise iff σ fixes c̄ pointwise.

(2) c̄ is a weak canonical base for ϕ(x̄, \overline{a}) if there are \overline{c}_{1} = \overline{c}, …, \overline{c}_{n} such that, for all automorphisms σ of the monster model of T, σ fixes A setwise iff it fixes \{c̄_{1}, …, \overline{c}_{n}\} setwise.

(3) T has (weak) elimination of imaginaries if every formula ϕ(x̄, \overline{a}) has a (weak) canonical base.

(4) T has p.p. elimination of imaginaries if every p.p. formula ϕ(x̄, \overline{a}) has a canonical base.

Equivalently ([9]), c̄ is a (weak) canonical base for ϕ(x̄, \overline{a}) if there is a formula ψ(x̄, ȳ) such that ∀x̄ ϕ(x̄, \overline{a}) ↔ ψ(x̄, \overline{c}) and there is exactly one tuple (finitely many tuples) \overline{y} such that ∀x̄ ϕ(x̄, \overline{a}) ↔ ψ(x̄, \overline{y}).

In both cases, we say that the elimination is uniform if ψ depends only on ϕ and not on the parameters \overline{a}.

Remark 9.2. Notice that, up to definable closure, a canonical base, when it exists, is uniquely determined. Moreover, if c̄ is a canonical base for ϕ(x̄, \overline{a}), then c̄ ∈ dcl(\overline{a}). Now, by Example 3.8 in [15], no theory of infinite modules has elimination of imaginaries.

Proposition 9.3. T_{e} (respectively T_{∞}) has uniform p.p. elimination of imaginaries.

Proof: (See also Lemma 3.7 in [15]).

Let ϕ(x̄, \overline{a}) be a p.p. formula. By the positive quantifier elimination for T_{e} and the additional fact that a conjunction of atomic formulas is equivalent to one atomic formula, we have that this formula is equivalent to t_{1}(x̄) = t_{2}(ā), where the t_{1}, t_{2}’s are L-terms. Set c̄ = t_{2}(ā). Now, suppose that we have another parameter b̄ such that T_{e} ⊧ ϕ(x̄, ā) ↔ ϕ(x̄, b̄), then t_{2}(ā)=t_{2}(b̄), hence c̄ is a canonical parameter for ϕ(x̄, \overline{a}). This elimination is uniform since it only depends on the formula (and not on the parameters) (see [13] p. 157). □

Corollary 9.4. T_{e} (respectively T_{∞}) has uniform weak elimination of imaginaries.

Proof: Any definable A can be finitely written as

A = ⋃_{i=1}^{n_{0}}(A_{i} \ ⋃_{j=1}^{n_{i}}(A_{ij} \ ⋃_{k=1}^{n_{ij}}(A_{ijk} \ …)))

where any A_{ν} occurring is a p.p. definable coset (with parameters). This decomposition is unique when some additional conditions are required: if N := {ν ∈ ω^{<ω}; A_{ν} exists }, A_{ν} should be connected for ν ∈ N (or rather the corresponding subgroup - we use here that a definable group is connected-by-finite, see Proposition 6.1), for each i ∈ ω such that furthermore (ν, i) ∈ N, A_{(ν,i)} should be strictly included in A_{ν} and finally, for fixed ν, the A_{(ν,i)}’s should be disjoint. In this situation, an automorphism of the monster model fixes A setwise iff it fixes setwise the set of A_{i}’s, i ∈ n_{0}. It is clear, by induction on the maximal length of ν ∈ N, that each A_{ν} has a canonical base.

Suppose now that A is defined by the formula ϕ(x, a) and let us introduce formulas ϕ_{ν} defining the A_{ν}’s. The ϕ_{ν}’s are given by the Baur-Monk p.p. quantifier elimination result in modules and are therefore uniform in a. By positive quantifier elimination in our structures, ϕ_{ν} has the form t_{ν}(x) = a_{ν}, where t_{ν} is a term without parameters, not depending on a, and the formula t_{ν}(x) = 0 defines a connected subgroup G_{ν}, strictly containing the group G_{(ν,i)} (by hypothesis on the decomposition), for any (ν, i) ∈ N. Hence the formula

⋁_{i=1}^{n_{0}}(ϕ_{i} ∧ ¬ ⋁_{j=1}^{n_{i}}(ϕ_{ij} ∧ ¬ ⋁_{k=1}^{n_{ij}}(ϕ_{ijk} ∧ ¬ …)))

provides a definition of A for only a finite choice of its parameters. □

In the finite imperfection case, we could have deduce weak elimination of imaginaries from our description of types, but a priori not the uniformity. Indeed, let us call canonical base of a type q over a substructure of a model M of T a definably closed set D ⊆ M such that, for every automorphism σ of the monster model of T, σ fixes q iff σ fixes D pointwise. As we see in the next lemma, in our theory, types do have canonical bases. Now Evans, Pillay and Poizat proved that a stable theory, where each n-type over every model has a canonical base, has weak elimination of imaginaries. (In fact, only a weaker property is required, namely to have a weak canonical base, whose definition we will not recall since in our case a stronger property holds).

Lemma 9.5. Let A be an L-substructure of a model of T_{e}. Then every type q ∈ S(A) has a canonical base D.

Proof: The map (x_{i})_{i∈p^{ek}} ↦ ∑_{i∈p^{ek}} x_{i} · t^{i} · m_{i} embeds S_{p^{ek}}(A) in S_{1}(A), so let us consider the case of 1-types. Let a ∈ M be a realization of q and let

D := {m ∈ A; x_{j_{0}} · r_{0} + … + x_{j_{s}} · r_{s} + m ∈ I(a_{∞}, A)}.

Note that D is a substructure of A. By Lemma 4.6, D is definably closed and by Lemma 5.7, it is clear that for every automorphism σ of M, σ fixes q iff σ fixes D pointwise. □

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Pilar Dellunde, Àrea de Lògica, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain

  E-mail address: Pilar.Dellunde@uab.es

Françoise Delon, C.N.R.S., U.M.R. 7056 et Université Paris 7, U.F.R. de Mathématiques, Case 7012, 2 place Jussieu, 75251 Paris Cedex 5, France

  E-mail address: delon@logique.jussieu.fr

Françoise Point, Institut de Mathématique et Informatique, Université de Mons-Hainaut, Le Pentagone, 6, avenue du Champ de Mars, B-7000 Mons, Belgium

  E-mail address: point@logique.jussieu.fr