THE THEORY OF MODULES OF SEPARABLY CLOSED FIELDS 1

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

Abstract. We consider separably closed fields of characteristic p > 0 and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the p-component functions.

1. Introduction

We will denote by SCF_{p,e} the first-order theory of separably closed fields of characteristic p > 0 and imperfection degree e ∈ ω ∪ {∞} in the language of fields. Y. Ershov showed that SCF_{p,e} is complete (see [5]). Moreover, whenever e is finite, if one adds new constants for the elements of a chosen p-basis and the p^{e} unary functions sending an element to its p-components over this basis, one gets quantifier elimination in this extended language (see for instance [3], Proposition 27).

Let K be a model of SCF_{p,e}; we will consider additive reducts of this field. This kind of structures was first considered in the light of Zil’ber conjecture on the possible geometries on strongly minimal sets. In the case where K is algebraically closed and f(x) is any polynomial with coefficients in K, Gary Martin proved that either f(x) is affine or multiplication is definable in an expansion of (K, +, f) by finitely many scalar multiplications (see Theorem 3.7 in [7]). (A polynomial is additive if it is a linear combination of X^{q}’s, where q is a power of p and it is affine if it differs by a constant from an additive polynomial.) This specific result is of interest to us because its proof uses only the fact that K is separably closed.

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

Here, we investigate the first-order theory of K regarded as a module over the skew polynomial ring R = 𝔽_{p}(B)[t; α] (see [2]), where B is a p-basis of K (hence a set of cardinality e), 𝔽_{p}(B) is the fraction field of the polynomial ring 𝔽_{p}[B] and α the Frobenius map (i.e., p^{th} power). Note that, in the context of modules, the cardinality of B is important, also when infinite: it defines the language in which we work. We extract a series of properties of these fields when viewed as R-modules and we show that the corresponding theory of modules over R is model-complete (and decidable). In the resulting structures, the decomposition of an element over the p-basis can be expressed and we will extend the ordinary language of modules with the analog of the p-component functions. Let T_{e} denotes the theory in this expanded language of the structures we are considering. The analysis proceeds first in investigating the torsion part, which comes down to the question whether we can describe in that weaker language the behaviour of roots of additive polynomials. Second, we show that any positive primitive formula is equivalent to a positive quantifier-free formula. Then we show that the index of one p.p. definable subgroup in another is either 1 or ∞ in any torsion-free summand of a model of our theory. This suffices to prove on one hand that T_{e} admits quantifier elimination, and on the other hand that it is the theory of separably closed fields of characteristic p and imperfection degree e in that weaker language.

Date: March 12, 2002.

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

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

In a second article (see [4]), we will show that (as in the field case [11]) for e > 0 T_{e} is non-superstable (though stable) and we will give a partial description of the closed subset of the Ziegler spectrum corresponding to this theory. Then we will characterize the types as certain submodules, we will identify the types of (finite) U-rank and we will show that we don’t have elimination of imaginaries but p.p. elimination of imaginaries.

2. Preliminaries

Fix a prime number p and a cardinal number e.

Definition 1. Let 𝔽_{p}(B) be the fraction field of the polynomial ring 𝔽_{p}[B], where B = {b_{i} : i ∈ e}. Let R=𝔽_{p}(B)[t; α] be the skew polynomial ring in t over 𝔽_{p}(B) associated with the Frobenius map α (see [2]). The domain of R is the set of polynomials in t of the form q(t)= t^{n}.a_{n} +...+t.a_{1} +a_{0}, where the a_{i}’s belong to 𝔽_{p}(B), with multiplication defined by the commutation rule a.t = t.a^{α}, for any a ∈ 𝔽_{p}(B). If a_{n} is different from zero, we call n the degree deg(q) of q(t) and q is called monic if a_{n} equals 1. We also set deg(0) = −∞. We have the usual rule deg(q.r)= deg(q) + deg(r), and so the right Euclidean division algorithm holds.

Substituting x^{p^{i}} in q(t) for t^{i} for all i (see Remark 1, item 3), the resulting ordinary polynomials are exactly the additive polynomials. They have been first considered by O. Ore who called them p-polynomials (see [8]). Equipped with addition and composition, the set of additive polynomials forms a ring. As noted by Ore, an ordinary polynomial having only simple roots is an additive polynomial iff its roots form an 𝔽_{p}-vector space (see Theorem 8, p.565 in [8]). This entails that any polynomial divides (in the ordinary sense) some additive polynomial (see Chapter 3, p.581 in [8]).

Let us recall a few properties of this ring, which can be found in Chapter 2 of [2]. By Proposition 2.11 in [2], R is an integral domain where the right ideals are principal. The proof is straightforward using the right Euclidean division algorithm. This entails that this ring is right Ore (see Corollary 1.3.7 in [2]), but whenever e ≠ 0, it is not left Ore i.e. there exist two non zero elements x_{1} and x_{2} such that R.x_{1} ∩ R.x_{2}={0}. (For instance take the elements t and t.b, for some b ∈ B.) So, it contains the free algebra over 𝔽_{p} on two generators x_{1} and x_{2} (see Proposition 1.6.6 in [2]). We don’t know whether the theory T_{R} of all right R-modules is undecidable.

3. Axiomatization. Finite imperfection degree.

For S a ring, the language of right S-modules is L_{S} = {+, −, 0, ·r : r ∈ S}, where for any r ∈ S and x in a right S-module, x · r denotes the scalar multiplication of x by the ring element r. Let T_{S} be the theory of all right S-modules in this language.

We extend the usual R-module language by adding new unary functions which will be existentially L_{R}-definable in the theory we will be considering (in case e is finite, they will be definable by a positive primitive formula).

Let e be finite and let B={b_{0}, ⋯, b_{e−1}}.

Definition 2. Let L = L_{R} ∪ {λ_{i} : i ∈ p^{e}}, where the λ_{i}’s are unary functions.

Given i ∈ p^{e}, let m_{i} be the monomial b_{0}^{i(0)}. · · · .b_{e−1}^{i(e−1)} (we identify here p^{e} with the set of all maps from {0, ⋯, e − 1} to {0, ⋯, p − 1}). We will call the m_{i}’s p-monomials.

Definition 3. Let T_{e} be the following L-theory:

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

2. For each q ∈ R with q(0) ≠ 0, ∀x∃y, x = y · q,

3. ∃^{=d}x_{i} (the x_{i} ’s are linearly independent over 𝔽_{p} ∧ ⋀ x_{i} · q = 0), for each q ∈ R of degree d with q(0) ≠ 0,

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

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

In the sequel, we find it useful to express some of our results for subtheories of T_{e}, namely:

Notation 3.1. Let T_{sep} be the L_{R}-theory consisting of the axiom schemes 1,2,3 above. Let T_{free} be the L-theory consisting of the axiom schemes 1,2,4,5. Note that the class of models of T_{free} is closed under direct summands (in the class of L-structures) and direct products (the axioms are Horn sentences). Let T_{λ} be the L-theory consisting of the axiom schemes 1,4,5.

Remark 1. 1. Note that we could have written axiom schemes 2 and 3 in a shorter

way as : ∀x∃^{=p^{d}}y(x = y · q), for each q ∈ R of degree d with q(0) ≠ 0.

2. Observe that if M is a model of T_{λ}, we have

M = ⊕_{i∈p^{e}}M · t · m_{i},

as 𝔽_{p}-vector-spaces. Note that an L_{R}-substructure of a model of T_{λ} is itself a model of axioms 4,5 if and only if it is an L-substructure.

3. Let q be an element of R of the form q = t^{n}.a_{n} + ⋯ + t.a_{1} + a_{0}. We denote by q[x^{p}] the polynomial a_{n}x^{p^{n}} + ⋯ + a_{1}x^{p} + a_{0}x. We say that q is separable if q(0) ≠ 0. This terminology is justified by the fact that q is separable as defined above if and only if q[x^{p}] is separable in the classical sense, that is, its formal derivative is non-zero.

Proposition 3.1. T_{e} is consistent.

Proof: Let K be a separably closed field of characteristic p and imperfection degree e. We interpret B as a p-basis of K. We define the action of t on K as the Frobenius map and so for instance we get for y ∈ K and q ∈ R, y · q = q[y^{p}]. We interpret the unary functions λ_{i} as the p-components on B. Since B is a p-basis, K is a model of axioms 4 and 5 and since K is separably closed, it satisfies T_{sep} and so K is a model of T_{e}. □

Observe that with the same interpretation of the action of t and of the λ_{i}’s, any field extension of 𝔽_{p}(B) admitting B as a p-basis, is a model of T_{λ}.

Notation 3.2. 1. Given q ∈ R , we will define \sqrt[p]{q}. First, for a = ∑_{i∈p^{e}} a_{i}^{p}.m_{i} ∈ 𝔽_{p}(B), where the a_{i}’s belong to 𝔽_{p}(B), set a^{1/p} := ∑_{i∈p^{e}} a_{i}.m_{i}. (Observe that (a^{p})^{1/p} = a, but except if a ∈ 𝔽_{p}(B)^{p}, (a^{1/p})^{p} and a are distinct.) Then, for q = ∑_{i=0}^{n} t^{j}.a_{j} ∈ R with a_{j} ∈ 𝔽_{p}(B), set \sqrt[p]{q} := ∑_{i=0}^{n} t^{j}.a_{j}^{1/p}. Iteration m times of ^{p} √ is denoted ^{p^{m}} √ .

2. Let q = ∑_{j=0}^{n} t^{j}.a_{j} ∈ R with a_{j} = ∑_{i∈p^{e}} a_{ji}^{p}.m_{i}, a_{ji} ∈ 𝔽_{p}(B). For i ∈ p^{e}, set

q_{(i)} := ∑_{j=0}^{n} t^{j}.a_{ji}^{p}. So, q = ∑_{i∈p^{e}} q_{(i)}.m_{i}.

3. Any nonzero q ∈ R can be written q = t^{m}.(t^{n}.a_{n} + ... + t.a_{1} + a_{0}) ∈ R, with a_{0}, …, a_{n} ∈ 𝔽_{p}(B) and a_{n}.a_{0} ≠ 0. Define v(q) = m and write q = t^{v(q)}.q_{s}. Note that q_{s} is separable.

Remark 2. Let q ∈ R. Then 1. t.q_{(i)} = \sqrt[p]{q_{(i)}}.t, for each i ∈ p^{e}.

2. If q is separable, then for some i ∈ p^{e}, \sqrt[p]{q_{(i)}} is also separable.

3. t.q = ∑_{i∈p^{e}} \sqrt[p]{q_{(i)}}.t.m_{i}.

We state now some preliminary lemmas on the L-terms. First, we introduce a notation for the composite of the functions λ_{i}.

Notation 3.3. For any i_{1}, …, i_{m} ∈ p^{e}, let λ_{i_{1},... ,i_{m}} := λ_{i_{m}} ∘ ⋯ ∘ λ_{i_{1}}. Each x in some model M of T_{λ} can be written uniquely as

x = ∑_{i_{1},... ,i_{m}∈p^{e}} λ_{i_{m}}(⋯ (λ_{i_{1}}(x)) ⋯) · t^{m} · m_{i_{m}}^{p^{m−1}}. · · · .m_{i_{1}}.

We will index by p^{em} = p^{me}=(p^{m})^{e} the p-components iterated m times and letting

m_{(i_{1},··· ,i_{m})} = m_{i_{m}}^{p^{m−1}}. · · · .m_{i_{1}} = b_{0}^{i_{1}(0)+i_{2}(0).p+···+i_{m}(0).p^{m−1}}. · · · .b_{e−1}^{i_{1}(e−1)+i_{2}(e−1).p+···+i_{m}(e−1).p^{m−1}},

we get x = ∑_{k∈p^{m}^{e}} λ_{k}(x) · t^{m} · m_{k}. We extend our notation to write an element q ∈ R as: q = ∑_{k∈p^{me}} q_{(k)}.m_{k}. Iterating the above Remark, we get that t^{m}.q = ∑_{k∈p^{me}} \sqrt[p^{m}]{q_{(k)}}.t^{m}.m_{k}.

Lemma 3.2. Let M be a model of T_{λ}. Then, 1. For any u, v ∈ M and i ∈ p^{e}, λ_{i}(u + v) = λ_{i}(u) + λ_{i}(v).

2. For any u ∈ M, d = ∑_{s∈p^{e}} d_{s}^{p}.m_{s} ∈ 𝔽_{p}(B) and i ∈ p^{e},

λ_{i}(u · d) = ∑_{δ∈2^{e}}( ∑_{\begin{matrix}r,s∈p^{e} \\ r+s=i+pδ \end{matrix}} λ_{r}(u) · d_{s} · ∏_{j∈e} b_{j}^{δ(j)}).

3. For any u ∈ M, d = ∑_{s∈p^{e}} d_{s}^{p}.m_{s} ∈ 𝔽_{p}(B), k an integer ≥ 1 and i ∈ p^{e},

λ_{i}(u · t^{k} · d) = u · t^{k−1} · d_{i}.

Proof: Immediate calculation from the definition of the functions λ_{i}. □

The above Lemma shows in fact that any model of T_{λ} is a module over the ring R[λ_{i} : i ∈ p^{e}] where the multiplication rules are given by 2. and 3. above (see [1]).

Corollary 3.3. Let M be a model of T_{λ}. Then any L-term t(x_{0}, …, x_{k}) is in M equal to a term of the form

∑_{j=0}^{k} ∑_{i∈p^{em}} λ_{i}(x_{j}) · d_{ji}

for some integer m ≥ 1 and d_{ji}’s in R.

Proof: Using Lemma 3.2 we show, by induction on the length m of the tuple j ∈ p^{em}, that λ_{j}(x · r) = ∑_{i∈p^{em}} λ_{i}(x) · r_{i}, for some r_{i}’s ∈ R. □

Lemma 3.4. Let m ∈ ℕ, m > 0, and q_{j}, q′_{j} ∈ R, with q_{j} = t^{m}.q′_{j}. Then the equation ∑_{j} y_{j} · q_{j} = u is equivalent to

⋀_{i∈p^{em}} ∑_{j} y_{j} · \sqrt[p^{m}]{(q′_{j})_{(i)}} = λ_{i}(u)

in any model M of T_{λ}.

Proof: By Remark 2 and Notation 3.3, q′_{j} = ∑_{i∈p^{em}}(q′_{j})_{(i)}.m_{i} and

∑_{j} y_{j} · t^{m} · q′_{j} = ∑_{i∈p^{em}}(∑_{j} y_{j} · \sqrt[p^{m}]{(q′_{j})_{(i)}}) · t^{m} · m_{i}.

Therefore, if ∑_{j} y_{j} · q_{j} = u, since M is model of T_{λ}, we have

⋀_{i∈p^{em}} ∑_{j} y_{j} · \sqrt[p^{m}]{(q′_{j})_{(i)}} = λ_{i}(u).

The converse is clear. □

Proposition 3.5. Let M be a model of T_{λ}, then the torsion part M_{tor} of M is:

{x ∈ M : x · q = 0, for some separable q ∈ R}

and M_{tor} is an L-substructure of M. Moreover, if M is a model of T_{free} (respectively T_{e}), then M_{tor} is a model of T_{free} (respectively T_{e}).

Proof:

1. Since R is a right Ore ring, the set of torsion elements forms a submodule. Let x_{1} and x_{2} belong to M_{tor} with x_{1}·q_{1} = 0, x_{2}·q_{2} = 0, q_{1}, q_{2} ∈ R−{0}. Then there exist a_{1}, a_{2} ∈ R−{0} such that q_{1}.a_{1} = q_{2}.a_{2} = q ≠ 0 and so (x_{1}+x_{2})·q = 0. For r ∈ R, let s = lcm(r, q_{1}) = r.r_{1} for some r_{1} ∈ R − {0}, then x_{1} · r · r_{1} = x_{1} · s = 0 and so x_{1} · r belongs to M_{tor}.

2. Assume that x ∈ M_{tor} with x · q = 0, for some q ∈ R − {0} with v(q) > 0. Let q = t^{v(q)}.q_{s}. By Remark 2(3) and axioms 4,5, ⋀_{i∈p^{e}} x · t^{v(q)−1} · \sqrt[p]{(q_{s})_{(i)}} = 0. Now, since q_{s} is separable, by Remark 2(2), for some i_{0}, \sqrt[p]{(q_{s})_{(i_{0})}} is separable. By iterating this procedure we obtain x · q′ = 0, for some q′ separable.

3. Let q = ∑_{i} t^{i}.a_{i} be a separable element of R and suppose that x · q = 0. Then writing q.a_{0}^{−1} = 1 + t.q′, we get x = − ∑_{i∈p^{e}} x · \sqrt[p]{q′_{(i)}} · t · m_{i}, so λ_{i}(x) = x · \sqrt[p]{q′_{(i)}} belongs to the R-submodule generated by x.

4. So, M_{tor} is an L-substructure of M. To check that it is a model of T_{free} (respectively T_{e}) whenever M is, in both cases, it is enough to check axiom scheme 2. Let x ∈ M_{tor}, q ∈ R − {0} and y such that x = y · q. Since R is a domain, y belongs to M_{tor}.

□

Corollary 3.6. Let K be a separably closed field with p-basis B. Let us denote by 𝔽_{p}(B)^{sep} the separable closure of 𝔽_{p}(B). Then K_{tor} = 𝔽_{p}(B)^{sep}.

Proof: Since any polynomial divides an additive polynomial (see [8] Chapter 3), we have that 𝔽_{p}(B)^{sep} ⊆ K_{tor}. Now K_{tor} ⊆ 𝔽_{p}(B)^{sep} since Proposition 3.5 tells us that x ∈ K_{tor} implies that x · q = 0 for some separable q. □

Remark 3. The proof of Proposition 3.5 may be generalized as follows. Let A, C be two L-structures models of T_{λ} and suppose that A ⊆ C. Let u belong to C and assume that there exists q ∈ R − {0} such that u.q belongs to A. Then there exists a separable element r ∈ R such that u · r belongs to A.

4. Axiomatization. Infinite imperfection degree.

In this section, we will extend our previous results to the infinite imperfection case.

Let B be infinite. Let L_{∞} = L_{R} ∪ {λb̄ ; i : i ∈ p^{n}, b̄ ∈ B^{n}, n ∈ ω − {0}}, where the λb̄ ; i’s

are unary functions. We fix an enumeration m_{n,i}(x̄) of the monomials of the form x_{0}^{i(0)} ⋯ x_{n−1}^{i(n−1)}, with n ∈ ℕ and i ∈ p^{n}. As in the finite imperfection degree case, we will call the elements of the form m_{n,i}(b̄) where b̄ ∈ B^{n}, p-monomials.

Definition 4. Let T_{∞} be the axiom schemes 1,2,3 as in the finite imperfection degree case together with the following schemes:

4_{∞}. ∀x( ⋁_{i∈p^{n}} λb̄ ; i(x) ≠ 0 → x = ∑_{i∈p^{n}} λb̄ ; i(x) · t · m_{n,i}(b̄)), for each b̄ = (b_{0}, …, b_{n−1}) ∈

B^{n}, n ∈ ℕ − {0}.

5_{∞}. ∀x∀(x_{i})_{i∈p^{n}}(x = ∑_{i∈p^{n}} x_{i}·t·m_{n,i}(b̄) → ⋀_{i∈p^{n}} x_{i} = λb̄ ; i(x), for each b̄ = (b_{0}, …, b_{n−1}) ∈

B^{n}, n ∈ ℕ − {0}.

A model of T_{∞} is K_{0} = 𝔽_{p}(B)^{sep}, where the functions λb̄ ; i are interpreted as follows. If x ∉ K_{0}^{p}(b̄), then λb̄ ; i(x) = 0, for any i ∈ p^{n}. For x = ∑_{i} x_{i}^{p}.m_{n,i}(b̄) for some x_{i}’s in K_{0}, then λb̄ ; i(x) = x_{i}. So, T_{∞} is consistent.

Similarly to the finite imperfection case, we will denote by T_{λ,∞} the L_{∞}-theory consisting of axiom scheme 1 together with axiom schemes 4_{∞} and 5_{∞} and by T_{free,∞} the L_{∞}-theory consisting of T_{λ,∞} together with axiom scheme 2. Observe that if M is a model of T_{λ,∞}, then for any integer n and b̄ ∈ B^{n} the sum of the 𝔽_{p}-vector spaces M · t · m_{n,i}(b̄) in M is direct. And, for x ∈ M, λb̄ ; i(x) ≠ 0 for some b̄ ∈ B^{n} and some i ∈ p^{n} if and only if x is a nonzero element of the direct sum ⊕_{i∈p^{n}}M · t · m_{n,i}(b̄) for some b̄ ∈ B^{n}.

In order to extend Notation 3.2 to the case when e is infinite, note that each q ∈ R belongs to 𝔽_{p}(b̄)[t; α] for some n ∈ ℕ and b̄ ∈ B^{n}. Then we obtain analogous statements to Remark 2 and Proposition 3.5.

Now we need to introduce a ”tree property” which indicates for a specific element the width of the tree of its (non-trivial) p-components.

Notation 4.1. For any sequences b̄ ∈ B^{n} and ī = (i_{1}, …, i_{m}) ∈ p^{n} × ⋯ × p^{n}, set λb̄ ; ī := λb̄ ; i_{m} ∘ … ∘ λb̄ ; i_{1}. As in Notation 3.3, we will identify the set (p^{n})^{m} with the set of maps from n to p^{m}, and for i such map, define m_{n,i}(b̄) := ∏_{j∈n} b_{j}^{i(j)}). From now on, we drop the “bar” above the index “i”.

Definition 5. Let M be an L_{∞}-structure, u ∈ M, b̄ ∈ B^{n}, m an integer ≥ 1. We say that u has the (b̄, m)-tree property (t.p.): if u = ∑_{i∈p^{nm}} λb̄ ; i(u) · t^{m} · m_{n,i}(b̄). Note that this property is expressible by a positive quantifier-free formula.

Remark 4. Let M be a model of T_{λ,∞}, u ∈ M, b̄ = (b_{0}, …, b_{n−1}) ∈ B^{n}, m an integer ≥ 1. The following are equivalent:

1. u has the (b̄, m + 1)-t.p.

2. u has the (b̄, 1)-t.p. and for any k ≤ m, any i =(i_{1}, …, i_{k}), λb̄ ; i(u) has the (b̄, 1)-t.p.

3. u has the (b̄, 1)-t.p. and for any i ∈ p^{n}, λb̄ ; i(u) has the (b̄, m)-t.p.

The proofs of the next three Lemmas are immediate using the definition of the functions λb̄ ; i.

Lemma 4.1. Let M be a model of T_{λ,∞}, u, v ∈ M and d ∈ 𝔽_{p}(B), d = ∑_{i∈p^{n}} d_{i}^{p}.m_{n,i}(b̄), for some b̄ = (b_{0}, …, b_{n−1}) ∈ B^{n} and d_{i} ’s in 𝔽_{p}(B). Then,

1. If u and v have the (b̄, 1)-tp, then for any i ∈ p^{n},

λb̄ ; i(u + v) = λb̄ ; i(u) + λb̄ ; i(v).

2. For any i ∈ p^{n}, k integer ≥ 1,

λb̄ ; i(u · t^{k} · d) = u · t^{k−1} · d_{i}.

3. For any i ∈ p^{n},

λb̄ ; i(u · d) = ∑_{δ∈2^{n}} ∑_{\begin{matrix}r,s∈p^{n} \\ r+s=i+pδ \end{matrix}} λb̄ ; r(u) · d_{s} · ∏_{j∈n} b_{j}^{δ(j)}

□

Lemma 4.2. Let M be a model of T_{λ,∞} and u ∈ M. Let b̄ ∈ B^{n} and q ∈ 𝔽_{p}(b̄)[t; α] be separable. Then, for any positive integer m, u has the (b̄, m)-t.p. iff u · q has the (b̄, m)-t.p. □

Lemma 4.3. Let q_{j}, q′_{j} ∈ R, q_{j} = t^{m}.q′_{j} where m > 0. Assume that the q_{j}’s belong to 𝔽_{p}(b̄)[t; α] where b̄ ∈ B^{n}. Then the equation ∑_{j} y_{j} · q_{j} = u is equivalent to

⋀_{i∈p^{nm}} ∑_{j} y_{j} · \sqrt[p^{m}]{(q′_{j})_{(i)}} = λb̄ ; i(u) and (u has the (b̄, m)-t.p. property)

in any model M of T_{λ,∞}. □

5. The Torsion part and divisible closure over a substructure.

The main part of this section will be devoted to the proof that the torsion submodules of any two models of T_{e} are isomorphic. Then we will see that we can adapt our proofs to show that any L-substructure of a model of T_{e} can be extended in a canonical way to a submodel.

Definition 6. Let q be an element of R, q is said to be prime if it is non invertible, separable and if it cannot be written as a product of two non invertible elements of R.

Fix an integer n ≥ 1. Let N be an L_{R}-structure, N_{0} a substructure of N and u ∈ N − N_{0} be such that u · q = 0, for some separable q ∈ R of degree n.

Lemma 5.1. There is a monic separable polynomial q_{u} ∈ R with degree ≤ n such that:

1. q_{u} is of degree minimal such that q_{u} ≠ 0 and u · q_{u} ∈ N_{0};

2. For any r ∈ R such that u · r ∈ N_{0}, q_{u} divides r on the right. Therefore, there is an unique monic separable polynomial q_{u} ∈ R of minimal degree such that u · q_{u} ∈ N_{0}. Suppose futhermore that N_{0} contains all elements of N_{tor} annihilated by some separable element of R of degree < n, then q_{u} is prime.

Proof: Let I be the set of polynomials r such that u · r ∈ N_{0}. By hypothesis, I contains a separable polynomial q. Since N_{0} is a right R-module, I is a right ideal and since R is right principal, there is q_{u} a monic polynomial generating I, moreover such polynomial is unique. Since q is separable, q_{u} is separable.

The last assertion under the additional hypothesis on N_{0} is clear. □

For c ∈ N, or c ⊆ N, we denote by 𝔽_{p}(B)[c] the 𝔽_{p}(B)-vector subspace of N generated by c and we denote by N_{0}⟨c⟩_{R} (respectively ⟨c⟩_{R}) the R-submodule of N generated by N_{0} ∪ {c} or N_{0} ∪ c (respectively c).

Corollary 5.2. Let u ∈ N −N_{0}, as above, and let n_{1} be the degree of q_{u}, then N_{0}⟨u⟩_{R} =N_{0}⊕𝔽_{p}(B)[u]⊕𝔽_{p}(B)[u·t]⊕⋯⊕𝔽_{p}(B)[u·t^{n_{1}−1}] as 𝔽_{p}(B)-vector spaces, i.e. N_{0}⟨u⟩_{R} is isomorphic to the direct sum of N_{0} and n_{1} copies of 𝔽_{p}(B).

Proof: Since u · q_{u} ∈ N_{0} and q_{u} is of degree minimal such, one has N_{0}⟨u⟩_{R} = N_{0} + 𝔽_{p}(B)[u, u · t, ..., u · t^{n_{1}−1}] and 𝔽_{p}(B)[u · t^{d}] ∩ (N_{0} + 𝔽_{p}(B)[u, u · t, ..., u · t^{d−1}]) = {0} for any d < n_{1}. □

Let N, M be two R-modules, models of T_{sep}, and suppose that N_{0} and M_{0} are two isomorphic submodules of N and M respectively which we assume to contain all the elements (in respectively N, M) annihilated by some separable element of R of degree < n. We denote by j this isomorphism and we want to extend it. Let u ∈ N_{tor} − N_{0} be such that u · q = 0, for some separable q ∈ R of degree n. Let q_{u} be the monic prime polynomial of degree ≤ n of Lemma 5.1. Note that u · q_{u} belongs to N_{tor} (N_{tor} is a submodule). Take \overline{u} to be an element of M such that \overline{u} · q_{u} = j(u · q_{u}). The existence of such an element is insured by axiom schemes 2 of T_{sep}. Moreover, since j(u · q_{u}) belongs to M_{tor}, \overline{u} also belongs to M_{tor}. Moreover, since N_{0} and M_{0} are isomorphic, q_{u} has the same number of roots in each of them, and since M and N are models of axiom schemes 3 of T_{sep}, q_{u} has a root in N − N_{0} iff it has one in M − M_{0}. So, we may choose \overline{u} in M_{tor} − M_{0} and note that since q_{u} is prime, it is of minimal degree such that \overline{u} · q_{u} ∈ M_{0}.

Lemma 5.3. The mapping u_{0} + u · r → j(u_{0}) + \overline{u} · r, for u_{0} ∈ N_{0} and r ∈ R, is well defined and is an isomorphism of R-modules between N_{0}⟨u⟩_{R} and M_{0}⟨\overline{u}⟩_{R} .

Proof: We will show only that it is well-defined. Using analogous arguments one can show that it is an isomorphism. Suppose that u_{0} + u · r = u′_{0} + u · r′, we have u · (r − r′) ∈ N_{0}. Therefore, by Lemma 5.1, r − r′ = q_{u}.s, for some s ∈ R. Then, 0 = (u_{0} − u′_{0}) + u · (q_{u}.s) and, since j is an isomorphism and by choice of ū,

0 = j(u_{0} − u′_{0} + u · q_{u} · s) = j(u_{0}) − j(u′_{0}) + j(u · q_{u}) · s
= j(u_{0}) − j(u′_{0}) + \overline{u} · q_{u} · s = j(u_{0}) − j(u′_{0}) + \overline{u} · (r − r′).

Consequently j(u_{0}) + \overline{u} · r = j(u′_{0}) + \overline{u} · r′. □

Notation 5.1. Let N be an R-module. Let N_{tor,sep}= {n ∈ N : n·q = 0 for some q ∈ R, q(0) ≠ 0}.

Proposition 5.4. Let N be any model of T_{sep}. Then N_{tor,sep} is isomorphic as an Rmodule to 𝔽_{p}(B)^{sep}. Given two models N, M of T_{sep} with N ⊆ M, we have N_{tor,sep} = M_{tor,sep}.

Proof: By induction we build two chains of submodules of N_{tor,sep} and 𝔽_{p}(B)^{sep} respectively, (N_{l} : l ∈ ω) and (M_{l} : l ∈ ω), such that for any l ∈ ω, N_{l} and M_{l} contains all the elements of respectively N and 𝔽_{p}(B)^{sep} annihilated by some separable element of R of degree < l. We define simultaneously a chain of isomorphisms (h_{l} : l ∈ ω), h_{l} : N_{l} → M_{l}. Take N_{0} = M_{0} = {0} and h_{0} the identity. Suppose inductively that we have defined N_{l}, M_{l} and h_{l}. Let (u_{i} : i ∈ ω) be an enumeration of the elements of N_{tor} annihilated by the separable elements of R of degree l. Using 5.3 we build by induction two chains of submodules of N_{tor,sep} and 𝔽_{p}(B)^{sep} respectively, (N_{l+1}^{i} : i ∈ ω) and (M_{l+1}^{i} : i ∈ ω), and a chain of isomorphisms (h_{l+1}^{i} : i ∈ ω) such that N_{l+1}^{0} = N_{l}, M_{l+1}^{0} = M_{l}, h_{l+1}^{0} = h_{l} and for any i ∈ ω, h_{l+1}^{i} : N_{l+1}^{i} → M_{l+1}^{i}, u_{i} ∈ N_{l+1}^{i+1}. Let N_{l+1}, M_{l+1} and h_{l+1} be the union of these chains. Since N is a model of axiom schemes 2, 3 of T_{sep}, it is easy to check that N_{tor,sep} = ⋃_{l∈ω} N_{l}, 𝔽_{p}(B)^{sep} = ⋃_{l∈ω} M_{l} and ⋃_{l∈ω} h_{l} : N_{tor,sep} → 𝔽_{p}(B)^{sep} is an isomorphism.

The number of elements in a model annihilated by a given separable element in R is finite and described by T_{sep}; this also gives the second assertion of the statement. □

Corollary 5.5. Let N be a model of T_{e}. Then, N_{tor} ≃ 𝔽_{p}(B)^{sep}.

Proof: By Proposition 3.5, we have that N_{tor}=N_{tor,sep}. So, we may apply the preceeding proposition. □

Let P be the set of elements of R corresponding to primitive additive polynomials (see [8]) w.r. to 𝔽_{p}(B)^{sep}; i.e., q ∈ P iff q(0) ≠ 0 and there is an element u ∈ 𝔽_{p}(B)^{sep} such that u · q = 0 but, for all polynomial r[x^{p}] properly dividing q[x^{p}] in the usual sense, one has u · r ≠ 0. Such an element u is called a primitive root of q. Note that O. Ore (see p. 582 in [2]) asked for necessary and sufficient conditions for an additive polynomial to have primitive roots. In particular, he gave an example of an additive polynomial without any.

Corollary 5.6. Let q be a separable element of R and let the r_{i}[x^{p}]’s to be all monic additive polynomials properly dividing q[x^{p}]. Then q belongs to P iff T_{sep} ⊢ ∃x(x · q = 0 ∧ ⋀_{i} x · r_{i} ≠ 0). □

Now we want to generalize Proposition 5.5 to the following situation. Let A be an L_{R}-substructure (respectively L-substructure) of a model M of T_{sep} (respectively T_{e}).

We will show in both cases that there is a canonical extension of A to a model of T_{sep} (respectively T_{e}). Even though in the first case it will be an L_{R}-structure and the second case an L-structure, the domain of both extensions will be the same and so we will denote it by the same letter.

By abuse of language, one might say that Proposition 5.4 was the case A = {0}. We need the following Lemma.

Lemma 5.7. The set R_{0} of separable elements of the skew polynomial ring R is a right denominator set; i.e., ∀r ∈ R, ∀r_{0} ∈ R_{0}, r.R_{0} ∩ r_{0}.R ≠ ∅.

Proof: Given two elements r and q of R, O. Ore constructs explicitly their least common multiple (as in the commutative case) (see [9]). One then checks that if r is separable, then lcm(r, q) = r.r_{1} = q.q_{1} with r_{1}, q_{1} ∈ R and q_{1} separable. □

Proposition 5.8. Let M, N be models of T_{sep} containing, respectively, isomorphic L_{R}-substructures A, B. Then we may extend this partial isomorphism to a minimal submodel A^{sep} of M containing A.

Proof: By Proposition 5.5, we know that the separable torsion submodules of M and N are isomorphic. Since A_{tor,sep}=M_{tor,sep} ∩ A, we may extend this isomorphism to an isomorphism from A + M_{tor,sep} to B + N_{tor,sep}.

To get a model of T_{sep}, we need to close these substructures by adding solutions of equations of the form (*) u · s = a, where a ∈ A and s is a separable element of R. Setting A_{0}=A + M_{tor}, then A_{1} is constructed solving equations of the form (*) with a ∈ A_{0}.

Let ϕ be the isomorphism between A_{0} and B_{0}, and suppose that there exists an element a in A_{0} which is not divisible in A_{0} by a separable polynomial, say q. Since M is a model of T_{sep}, there exists u belonging to M such that u · q = a. Let now s′ be a separable polynomial of degree minimal such that u · s′ belongs to A_{0}. Let a′ = u·s′. The set of solutions of the equation a′ = x·s′ is equal to u+{x : x·s′ = 0}, which means that A_{0} contains none of them. Let v ∈ N be such that v · s′ = ϕ(a′). Since A_{0} is isomorphic to B_{0}, v does not belong to B_{0}. Claim: s′ is of degree minimal such that v · s′ belongs to B_{0}. Proof: Let r be an element of R of degree minimal such that v · r belongs to B_{0}. Then r divides s′ i.e. there exists r_{1} ∈ R such that s′ = r.r_{1}. Set b_{0} = v · r. Since ϕ is an isomorphism between A_{0} and B_{0}, there exists c_{0} in A_{0} with ϕ(c_{0}) = b_{0}. Now a′ = u · s′ = u · r · r_{1} = c_{0} · r_{1}. So, u · r − c_{0} belongs to ker(r_{1}) which is included in A_{0}. Therefore, u · r belongs to A_{0} and by choice of s′, deg(r)=deg(s′). □

Extend ϕ on ⟨u⟩_{R}+A_{0} by setting ϕ̄(u·r+a_{0})=v·r+ϕ(a_{0}), where r belongs to R and a_{0} to A_{0}. This is well defined: suppose that u·r+a_{0} = u·r′ +a_{1} i.e. u·(r−r′) belongs to A_{0}. Then s′ divides r−r′ i.e. (r−r′) = s′.s_{1}. Therefore, u·(r−r′) = a′·s_{1} = a_{1}−a_{0}, so ϕ(a′ · s_{1}) = ϕ(a_{1} − a_{0}) and v · (r − r′) = v · s′ · s_{1} = ϕ(a′) · s_{1} = ϕ(a′ · s_{1}). Thus, v · (r − r′) = ϕ(a_{1}) − ϕ(a_{0}). This map is injective by the claim.

We repeat this process in order to get an R-module A_{1} in which all elements of A_{0} are divisible by each separable element of R. Claim: A_{1} is a model of T_{sep}. Proof: Let v ∈ A_{1} and u ∈ M be such that u · q = v, where q ∈ R and q(0) ≠ 0. We are going to show that u belongs to A_{1}. By construction of A_{1}, there exists u_{i}’s in M, r_{i}, q_{i}’s in R with q_{i}(0) ≠ 0 such that v = ∑_{i≥1} u_{i} · r_{i} and u_{i} · q_{i} ∈ A_{0}. Since R is right Ore, there exist s_{1} ∈ R with s_{1}(0) ≠ 0 (see Lemma 5.7), t_{1} ∈ R − {0} such that r_{1}.s_{1} = q_{1}.t_{1}. So we have v · s_{1}=∑_{i≥1} u_{i} · r_{i} · s_{1}=u_{1} · q_{1} · t_{1} + ∑_{i≥2} u_{i} · r_{i} · s_{1}. Iterating, we get first that there exist s_{2}, t_{2} ∈ R − {0} with s_{2}(0) ≠ 0 such that r_{2}.s_{1}.s_{2} = q_{2}.t_{2}. So we get v · s_{1} · s_{2}= a + ∑_{i≥3} u_{i} · r_{i} · s_{1} · s_{2}, where a ∈ A_{0}. Finally we get that there exists s ∈ R with s(0) ≠ 0 such that v · s ∈ A_{0}. So u · q · s ∈ A_{0}, which implies that u ∈ A_{1}. □

Corollary 5.9. Let M, N be two models of T_{e} containing two isomorphic L-substructures A, B, respectively. Then we may extend this partial isomorphism to A^{sep} which is a minimal submodel of M containing A.

Proof: We keep the same notations as in the proof above. We assume now that A is an L-structure and M is a model of T_{e}. We first check that A_{1} is an L-structure: if u · s = a for some separable element s of R, then λ_{i}(u) belongs to the R-submodule generated by u and λ_{j}(a), see proof (item 3) of Proposition 3.5, which also implies that the isomorphism ϕ̄ above commutes with the functions λ_{i}’s. Thus A_{1}=A^{sep} is the required minimal submodel. □

Remark 5. Note that in the above Corollary, we could have replaced T_{e} by any extension of T_{free} which specifies for each separable polynomial of R the number of elements annihilated by it.

6. Quantifier elimination in the case of finite imperfection degree

First, we note that the proof of the Proposition in [6] p.176, adapts to right Euclidean rings. Let us recall the definition.

Definition 7. Let S be a domain, it is a right Euclidean ring if there exists a function δ from S to ℕ such that ∀p_{1}∀p_{2}∃q∃r(p_{1} = p_{2}.q + r with δ(r) < δ(p_{2})) (see [6] p.143).

Proposition 6.1. Let S be a right Euclidean ring and A be a matrix with coefficients in S. Then there exist invertible matrices P, Q (P with coefficients in {0, 1}), such that P.A.Q is lower triangular.

Proof: For convenience of the reader, let us indicate the main steps of the proof, which goes by induction on the number of lines of the n × m matrix A. As usual, a lower triangular matrix C is of the form C = (c_{ij}), where c_{ij} = 0 if i < j.

1. Let P_{ij}= 1-e_{ii}-e_{jj}+e_{ij}+e_{ji}, where 1 is the identity matrix and e_{ij} the matrix with 1 in position (i, j) and 0 in the other positions. Note that P_{ij} is invertible, P_{ij}^{2}=1.

2. Let T_{ij}(c)= 1+e_{ij}.c, where c ∈ S and i ≠ j. The inverse of T_{ij}(c) is T_{ij}(−c).

3. P_{ij}.A: exchange the lines i, j in A.

4. A.P_{ij}: exchange the columns i, j in A.

5. A.T_{ij}(c): the j^{th} column of A is replaced by the sum of the i^{th} column ×c of A and the j^{th} column of A.

If A is non zero, let a_{ij} be a non zero element of minimal δ in matrix A. In the product P_{i1}.A.P_{1j}, the element a_{ij} is in the (1, 1)-position. Thus, we may assume that the initial matrix A has the property that the element in the (1, 1)-position is of minimal δ. Then we will show that by multiplying P_{i1}.A.P_{1j} by matrices of the form P_{sr} and T_{hk}(c) (on the right), we will obtain a matrix which first line only consists of zero’s except at the (1, 1)-position. So let us consider an element a_{1k}, with k ≠ 0. Performing the Euclidean division, there exist q_{k}, r_{k} in S such that a_{1k} = a_{11}.q_{k} + r_{k} with δ(r_{k}) < δ(a_{11}). In the product P_{i1}.A.P_{1j}.T_{1k}(−q_{k}) the element at the (1, k)-position is r_{k}. Either it is equal to 0, so we go on applying the process above to another element on the first line, or one multiplies the matrix obtained so far by P_{1k}. Now the element at the (1, 1)-position is the (nonzero) element r_{k} with δ(r_{k}) < δ(a_{11}). Therefore, the process terminates after finitely many steps and finally one obtains a matrix A′ of the form

( \begin{matrix}a & ¯0 \\ . & B \end{matrix} )

with a a nonzero element and B a (n − 1) × (m − 1) matrix to which one applies the induction hypothesis. Note that A′ is equal to the product of A by invertible matrices of the form P_{ij} or T_{1k}(c), c ∈ S, and note that on the right we only multiply by matrices of the form P_{ij}. Note that the reasoning does not depend on whether a_{ij} is a non zero element of minimal δ within the i^{th} line, or in the whole matrix A. □

Definition 8. A lower triangular n × m-matrix of co-rank ℓ is of the form (A_{1}, 0) where A_{1} is a lower triangular n × k-matrix (k ≤ n, m) with only non zero elements on its diagonal and 0 is a zero n × (m − k)-matrix, with ℓ = n − k.

Propsition 6.1 may be reformulated as follows.

Corollary 6.2. Any n × m-matrix A ≠ 0 is equivalent to a lower triangular matrix of co-rank ℓ, with n − ℓ > 0.

□

Now we will apply the above proposition in our setting, namely to the skew polynomial ring R. We note the following.

Corollary 6.3. Suppose that some line of A contains a separable element of R, then the element d_{11} on the diagonal of the lower triangular matrix P.A.Q, given by Proposition 6.1, is separable.

Proof: It suffices to note that while performing the Euclidean algorithm: a_{1k} = a_{11}.q_{k} + r_{k}, if a_{11}(0) = 0 and a_{1k}(0) ≠ 0, then r_{k}(0) ≠ 0 and if a_{11}(0) ≠ 0, either r_{k} = 0 and the element at the (1, k)-position is zero, so we keep a_{11} which is separable, or we switch the positions of r_{k} and a_{11} and we are back to the first case whenever r_{k}(0) = 0. □

Definition 9. An n×m-matrix is lower triangular separable (l.t.s.) if it is of the form (A_{1}, 0) where A_{1} is a lower triangular n × k-matrix (k ≤ n, m) with only separable elements on its diagonal and 0 is a zero n × (m − k)-matrix.

Lemma 6.4. Let A be an n × k-matrix with coefficients in R. Then the system of equations ȳ · A = ū is equivalent in any model M of T_{λ}, to a system

ȳ · P · B = \overline{t(ū)},

where P is a permutation matrix, B is a lower triangular separable n×k′-matrix and \overline{t} is a tuple of L-terms.

Proof: We will prove the Lemma by induction on the number n of lines of matrix A. Let a_{hℓ} be a coefficient of minimal valuation in matrix A, say w, and assume it is strictly positive. Multiplying A on the left by the matrix P_{1h}, we get this element on the first line of the resulting product. Set z̄=ȳ · P_{1h}^{−1}. Now, ∑_{j} z_{j} · a_{jℓ} = u_{ℓ} is equivalent by Lemma 3.4 to

⋀_{i∈p^{ew}} λ_{i}(u_{ℓ}) = ∑_{j} z_{j} · \sqrt[p^{w}]{(a′_{jℓ})_{(i)}},

with a′_{jℓ} ∈ R defined by a_{jℓ} = t^{w}.a′_{jℓ}. We have a_{jℓ}=∑_{i} (a_{jℓ})_{(i)}.m_{i}= t^{w}. ∑_{i} (a′_{jℓ})_{(i)}.m_{i}=

∑_{i} \sqrt[p^{w}]{(a′_{jℓ})_{(i)}}.t^{w}.m_{i} and \sqrt[p^{w}]{(a′_{hℓ})_{(i)}} is separable for at least one i. This amounts to

replacing each column of (P_{1h}.A) by p^{ew} columns and u_{ℓ} by the p^{ew}-tuple λ_{∗}(u_{ℓ}) = (λ_{i}(u_{ℓ}))_{i∈p^{ew}}. The system z̄ · P_{1h} · A = ū is equivalent to z̄ · Ã = λ_{∗}(ū), where the matrix Ã has a separable element on its first line.

Now by the above Corollary, one obtains a matrix C̃ of the form

( \begin{matrix}a & ¯0 \\ ā & C \end{matrix} )

with a a separable element, by multiplying the matrix Ã on the right by matrices of the form P_{ij} and T_{ij}(c), for some c ∈ R (since the non zero element of Ã is already on the first line we do not need to multiply Ã on the left by some P_{ij}’s). Let Q be the product of these invertible matrices and set Q = (Q_{1}, Q_{2}), where Q_{1} is the first column of Q. So, the system ȳ · A = ū is equivalent to z̄ · C̃ = λ_{∗}(ū) · Q, which in turn is equivalent to

∑ z_{i} · a_{i} = λ_{∗}(ū) · Q_{1} ∧ (z_{2}, ⋯, z_{n}) · C = λ_{∗}(ū) · Q_{2},

where a_{1}=a, (a_{i})_{2≤i≤n} = ā and the matrix C has n − 1 lines. We apply the induction hypothesis to C. □

Corollary 6.5. Every p.p. L_{R}-formula is equivalent to a conjunction of atomic Lformulas modulo T_{free}.

Proof: Let ϕ(x_{1}, ..., x_{m}) be a p.p. formula of the form ∃y_{1}, ..., y_{k}(\overline{x} · B = \overline{y} · A), where A, B are non zero matrices with coefficients in R. By Lemma 6.4, there exist a permutation matrix P, an l.t.s. k × n-matrix Ã and n L-terms t_{i}, 1 ≤ i ≤ n, such that:

ϕ(\overline{x}) ↔ ∃y_{1}, ..., y_{k}(t_{1}(\overline{x}), ⋯, t_{n}(\overline{x})) = \overline{y} · P · Ã.

Set \overline{y} · P =ȳ′. Assume that matrix Ã is of co-rank ℓ and write it as Ã = (A_{1}, 0), where A_{1} is an l.t.s. k × (k − ℓ)-matrix.

We have

ϕ(\overline{x}) ↔ ∃y′_{1}, ..., y′_{k}(t_{1}(\overline{x}), ⋯, t_{n}(\overline{x})) = \overline{y}′ · Ã)
↔ ∃y′_{1}, ..., y′_{k}[(t_{1}(\overline{x}), ⋯, t_{k−ℓ}(\overline{x})) = \overline{y}′ · A_{1} ∧ (t_{k−ℓ+1}(\overline{x}), ⋯, t_{n}(\overline{x})) = ¯0]
↔ (t_{k−ℓ+1}(\overline{x}), ⋯, t_{n}(\overline{x})) = ¯0.
□

Corollary 6.6. Let M be a model of T_{free}. Then M_{tor} is a pure submodule.

Proof: We have shown (see Proposition 3.5) that M_{tor} is an L-substructure. So, we may apply the above Corollary. □

In the next Lemma, we will use the following terminology. An L_{R}-inequation in variables x_{1}, ⋯, x_{n} and parameters v̄ is a basic formula of the form ∑_{i} x_{i} · r_{i} ≠ t(v̄), where r_{i} ∈ R, 1 ≤ i ≤ n, and t is an L_{R}-term. We will say that this inequation is nontrivial if for some i, r_{i} ≠ 0.

Lemma 6.7. Let T be the theory of all torsion-free right R-modules satisfying axiom scheme 2. Let Σ(x̄, ū) be a system of equations in L_{R} of the form x̄·A = ū, where A is a non zero l.t.s. m×k-matrix of co-rank ℓ, and x̄ = (x_{1}, ..., x_{m}), ū = (u_{1}, ⋯, u_{m−ℓ}, ¯0) are variables. Then, for any system Γ(x̄, v̄) of L_{R}-inequations in x̄ and parameters v̄, there exists a system of L_{R}-inequations Γ′(x_{m−ℓ+1}, ..., x_{m}, ū, v̄) such that Σ(x̄, ū) ∧ Γ(x̄, v̄) is equivalent in T to Σ(x̄, ū)∧Γ′(x_{m−ℓ+1}, ..., x_{m}, ū, v̄), with the convention that if ℓ = 0, the system of inequations is of the form Γ′(ū, v̄).

Moreover, we can split the set Γ′ into two subsets, Γ′_{0}(x_{m−ℓ+1}, ..., x_{m}, ū, v̄) consisting of non-trivial inequations and Γ′_{1}(ū, v̄) of trivial ones such that for any model M of T, for any ā, c̄ in M satisfying Γ′_{1}(ā, c̄), Σ(x̄, ā) ∧ Γ(x̄, c̄) has realizations in any non zero R-submodule M_{0} of M containing ⟨ā⟩^{sep}.

Proof: We proceed by induction on m. Write the matrix A as (ā, A_{1}), where ā is the first column of A. We associate pairwise the first equation ∑_{i} x_{i} · a_{i} = u_{1} of Σ and the j^{th} inequation of Γ: ∑_{i} x_{i} ·b_{ij} ≠ t_{j}(v̄), where t_{j}(v̄) is an L_{R}-term. Since R is right Ore, for each j such that b_{1j} ≠ 0, there exist c_{j}, b_{j} ∈ R − {0} such that a_{1}.c_{j} = b_{1j}.b_{j}. The system:

\{ \begin{matrix}∑_{i} x_{i} · a_{i} · c_{j} = u_{1} · c_{j} \\ ∑_{i} x_{i} · b_{ij} · b_{j} ≠ t_{j}(v̄) · b_{j} \end{matrix}

is equivalent in any torsion-free module to

\{ \begin{matrix}∑_{i} x_{i} · a_{i} = u_{1} \\ ∑_{i>1} x_{i} · (b_{ij}.b_{j} − a_{i}.c_{j}) ≠ t_{j}(v̄) · b_{j} − u_{1} · c_{j}. \end{matrix}

Let us denote by Γ′′(x_{2}, ⋯, x_{m}, ū, v̄) the set of inequations so obtained together with all the inequations of Γ where the coefficient of x_{1} is equal to 0. Either k = 1 and so Γ′ := Γ′′ is the required set of inequations, or k ≥ 2, so we apply the induction hypothesis. The system (x_{2}, ⋯, x_{m}) · A_{1} = (u_{2}, ⋯, u_{m−ℓ}, ¯0) ∧ Γ′′(x_{2}, ⋯, x_{m}, ū, v̄) is equivalent to (x_{2}, ⋯, x_{m}) · A_{1} = (u_{2}, ⋯, u_{m−ℓ}, ¯0) ∧ Γ′(x_{m−ℓ+1}, ⋯, x_{m}, ū, v̄), for a system Γ′ of inequations. Finally we obtain that the system Σ ∧ Γ is equivalent to Σ ∧ Γ′.

Let Γ′_{1}(ū, v̄) consist of all the trivial inequations in Γ′ and Γ′_{0}(x_{m−ℓ+1}, ..., x_{m}, ū, v̄) of the other ones.

Let M be a model of T, let ā, c̄ be in M and assume that Γ′_{1}(ā, c̄) holds. We will construct a solution x̄ in M of Σ ∧ Γ. Let us index the elements of Γ′_{0} by J and let us rewrite the j^{th} inequation as ∑_{i≥m−ℓ+1} x_{i} · d_{ij} ≠ t_{j}(ū, v̄), where d_{ij} ∈ R and t_{j} is an L_{R}-term, and let i_{j} be the first i such that d_{ij} ≠ 0. Let m_{1} be the maximal such i_{j}, j ∈ J, and let J_{1} be the subset {j ∈ J : i_{j} = m_{1}} of J. If the set {i : m_{1} < i ≤ m} is not empty, choose arbitrarily in M_{0} the components x_{m_{1}+1}, ⋯, x_{m} of x̄. Then, choose x_{m_{1}} ∈ M_{0} satisfying the following inequations: ⋀_{j∈J_{1}} x_{m_{1}} · d_{m_{1}j} ≠ t_{j}(ū, v̄) − ∑_{i>m_{1}} x_{i} · d_{ij} (there exists such an x_{m_{1}} in M_{0}, since there are only finitely many forbidden choices and M_{0} is infinite). Then, let m_{2} be the maximal i_{j}, j ∈ J − J_{1}, and let J_{2} be the subset {j ∈ J − J_{1} : i_{j} = m_{2}} of J − J_{1}. If the set {i : m_{2} < i < m_{1}} is not empty, choose arbitrarily in M_{0} the components x_{m_{2}+1}, ⋯, x_{m_{1}−1} of x̄. Then, choose x_{m_{2}} ∈ M_{0} satisfying the following inequations: ⋀_{j∈J_{2}} x_{m_{2}} · d_{m_{2}j} ≠ t_{j}(ū, v̄) − ∑_{i>m_{2}} x_{i} · d_{ij}. Proceed similarly up to the (m + 1 − ℓ)^{th}-component of x̄, then use the system of equations x̄ · A = ā to find successively, using axiom scheme 2, the first m − ℓ components of x̄ in M_{0}. It is at this point that we use the fact that M_{0} contains ⟨ā⟩^{sep}. □

Lemma 6.8. Let ϕ, ψ be two p.p. L_{R}-formulas defining two subgroups with ψ implying ϕ in T_{R}. Suppose that there exists a torsion-free L-structure M modelling T_{free} with [ϕ(M) : ψ(M)] > 1. Then, for any non-trivial torsion-free L-structure N modelling T_{free}, [ϕ(N) : ψ(N)]=∞.

Proof: Let a ∈ M be such that (ϕ(a) ∧ ¬ψ(a)). By Corollary 6.5, there exist finitely many L-terms t_{k}(x) (respectively s_{l}(x)) such that ϕ(x) is equivalent modulo T_{free} to ⋀_{k} t_{k}(x) = 0 (respectively ψ(x) is equivalent (modulo T_{free}) to ⋀_{l∈L} s_{l}(x) = 0). So, ∃x ∈ M(ϕ(x) ∧ ¬ψ(x)) is equivalent to ⋁_{l∈L} ∃x ∈ M (⋀_{k} t_{k}(x) = 0 ∧ s_{l}(x) ≠ 0). There is some m ≥ 1 such that the terms t_{k}(x) and s_{l}(x) are modulo T_{free} of the form ∑_{j∈p^{me}} λ_{j}(x) · d_{j} with d_{j} ∈ R (see Corollary 3.3). Let us use new variables y_{j} for the λ_{j}(x)’s. Identify p^{me} with {0, ⋯, p^{me} − 1} and set ȳ = (y_{0}, ⋯, y_{p^{me}−1}), and let t′_{k}(ȳ) = ∑_{j} y_{j} · r_{kj} (respectively s′_{l}(ȳ) = ∑_{j} y_{j} · s_{lj}) be the terms obtained by substituting y_{j} for λ_{j}(x), from t_{k}(x) (respectively s_{l}(x)).

The system ⋀_{k} t′_{k}(ȳ) = 0 can be written as ȳ · A=0 and by Lemma 6.4, it is equivalent (modulo T_{free}) to the system ȳ · P · Ã=0, where the matrix P is a permutation matrix and Ã is an l.t.s. matrix of non zero co-rank ℓ since ϕ/ψ(M) > 1. For ease of notation, set n = p^{me}, z̄ = ȳ · P, and ∑_{0≤j<n} y_{j} · s_{lj} = ∑_{0≤j<n} z_{j} · s̃_{lj}, for some s̃_{lj}’s in R.

Now, the system S_{l}(z̄) below, which is assumed to be consistent for each l ∈ L:

\{ \begin{matrix}z̄ · Ã = 0 \\ ∑_{0≤j<n} z_{j} · s̃_{lj} ≠ 0, \end{matrix}

is equivalent by Lemma 6.7 to a system of the form:

\{ \begin{matrix}z̄ · Ã = 0 \\ ∑_{n−ℓ<j<n} z_{j} · s′_{lj} ≠ 0, \end{matrix}

where the inequation is non-trivial by consistency of ϕ ∧ ¬ψ.

Let now N be a torsion-free L-structure, modelling T_{free}. Suppose we have constructed d elements x(1), ..., x(d) satisfying ϕ and such that all x(i), x(i)−x(j), i ≠ j, do not satisfy ψ. Denote by v_{j}(i), 1 ≤ i ≤ d, the λ_{j}-components, 0 ≤ j ≤ n − 1, of x(i). We are looking for an element x(d + 1) satisfying ϕ and such that x(d + 1) and x(d + 1) − x(i), 1 ≤ i ≤ d do not satisfy ψ, i.e. x(d + 1) = ∑_{0≤j<n} y_{j} · t^{m} · m_{j} such that z̄:= (y_{j})_{0≤j<n} · P satisfies one of the following systems S_{l,f}(z̄, v̄(1), ⋯, v̄(d)) := {z̄ · Ã = 0} ∪ Γ_{l,f}(z̄, v̄(1), ⋯, v̄(d)), where l ∈ L, f = (f(1), ⋯, f(d)) ∈ L^{d} and Γ_{l,f}(z̄, v̄(1), ⋯, v̄(d)) consists of :

\{ \begin{matrix}∑_{n−ℓ<j<n} z_{j} · s′_{lj} ≠ 0 \\ ⋀_{1≤i≤d} ∑_{n−ℓ<j<n}(z_{j} − v_{j}(i)) · s′_{f(i)j} ≠ 0. \end{matrix}

It suffices to apply Lemma 6.7 with ā = ¯0 and c̄= (v̄(1), ⋯, v̄(d)); note that in this case Γ′_{1} is empty.

□

Proposition 6.9. The theory T_{e} admits quantifier elimination and it is complete.

Proof: First, any L-formula is equivalent to an L_{R}-formula in T_{λ}. Second, any theory of modules admits p.p. elimination i.e., every formula is equivalent to a boolean combination of positive primitive formulas modulo invariant sentences. An invariant sentence is one which specifies the index for one p.p. definable subgroup included in another (see [10], Chapter 2). By Corollary 6.5, it suffices to prove that the index (in ω ∪ {∞}) of two p.p. definable subgroups (with one included in the other) is the same in any model of T_{e}. Note that this last property implies that T_{e} is complete. We may always work in an ℵ_{1}-saturated model M of T_{e}. By Proposition 5.5, the torsion submodel M_{tor} is isomorphic to 𝔽_{p}(B)^{sep} and in a nonprincipal ultrapower of M, say M^{∗}, the corresponding ultrapower M_{tor}^{∗} of M_{tor} is a direct summand as a pure-injective pure submodule (by Corollary 6.6) and M^{∗}/M^{∗}_{tor} is a torsion-free non zero model of T_{free}. But by the Lemma above, the index of two p.p. definable subgroups (with one included in the other) is the same in any non-trivial torsion-free R-module purely embedded in a model of T_{e}. This suffices by [10] Lemma 2.23. □

Remark 6. 1. In fact, we proved that not only our theory T_{e} admits quantifier elimination but that any positive primitive L-formula is equivalent to a conjunction of atomic L-formulas. This last property appears in the theory of modules and is denoted by “elim-Q^{+}” (see [10] p.319). In particular, it means that the p.p. definable functions are in fact definable by a conjunction of atomic formulas.

2. The same proof also gives that the theory of the class of torsion-free models of T_{free} admits q.e. in L as well as any extension of T_{free} which specifies for each separable polynomial the finite number of elements it annihilates.

Corollary 6.10. The L_{R}-reduct of T_{e} is model-complete. It is axiomatized by T_{sep} and ∀x ∃!(x_{i})_{i∈p^{e}} x = ∑ x_{i} · t · m_{i}.

Proof: The functions λ_{i}’s are L_{R}-existentially definable in any model of T_{e} and T_{e} admits quantifier elimination in the language L by the above proposition. □

Corollary 6.11. T_{e} has a prime model.

Proof: Apply Propositions 5.4 and 6.9. □

Corollary 6.12. T_{e} is decidable.

Proof: T_{e} is complete and recursively axiomatizable (a “natural” enumeration of R provides a recursively enumerable axiomatisation of T_{e}). The result may also be deduced from the fact that T_{e} is the L-reduct of the theory of the separably closed field 𝔽_{p}(B)^{sep} which is decidable. □

7. Quantifier elimination in the case of infinite imperfection degree

As in the finite imperfection degree case, the torsion submodule of any model of T_{∞} is isomorphic to 𝔽_{p}(B)^{sep}.

Any L_{∞}-formula is equivalent to an L_{R}-formula in T_{∞} since the λb̄ ; i functions are positively existentially L_{R}-definable.

Lemma 7.1. Let A be a h × k-matrix with coefficients in 𝔽_{p}(b̄)[t; α], for some tuple b̄ ∈ B^{n}. Then, there are L_{∞}-terms (τ_{l}(.))_{l=1}^{s} of the form τ_{l}(.) = λb̄ ; i_{l,1} ∘ … ∘ λb̄ ; i_{l,r_{l}}(.), with i_{l,j} ∈ p^{nm_{l,j}} for all j = 1, …, r_{l}, a k′-tuple of L_{R}-terms \overline{t}, a permutation matrix P, an l.t.s. h × k′-matrix B, and integers g_{l}, 1 ≤ g_{l} ≤ k, such that the system of equations ȳ · A = ū is equivalent, modulo T_{free,∞}, to

ȳ · P^{−1} · B = \overline{t}((λb̄ ; i_{l,1} ∘ … ∘ λb̄ ; i_{l,r_{l}}(u_{g_{l}}))_{l=1}^{s})

∧ ⋀_{\begin{matrix}1≤l≤s \\ 1≤j≤r_{l} \end{matrix}} λb̄ ; i_{l,j+1} ∘ … ∘ λb̄ ; i_{l,r_{l}}(u_{g_{l}}) has the (b̄, m_{l,j})-t.p.

Proof: The proof is analogous to the proof of Lemma 6.4. Note that at each step, we look at a coefficient of minimal valuation and if v̄ · Q = w̄, where Q is an invertible matrix with coefficients in 𝔽_{p}(b̄)[t; α], and if each component w_{i} of w̄ has the (b̄, m)-tp property for some m, then this also holds for each component v_{i} of v̄. □

Proposition 7.2. Every primitive positive L_{R}-formula is equivalent, modulo T_{free,∞}, to a positive quantifier-free L_{∞}-formula.

Proof: The proof is analogous to the proof of Corollary 6.5 with the additional information that the obtained terms occuring in the conjunction of atomic formulas equivalent to a given p.p. formula have the (b̄, m)-tp.-property, for some m and tuple b̄ ∈ B^{n}, for some n. Note that this last property is expressible by a positive quantifier-free L_{∞}-formula (see Definition 5). □

Lemma 7.3. Let ϕ, ψ be two p.p. L_{R}-formulas respectively defining two subgroups with ψ implying ϕ in T_{R}. Suppose that there exists a torsion-free L_{∞}-structure M modelling T_{free,∞}. with [ϕ(M) : ψ(M)] > 1, then [ϕ(N) : ψ(N)]= ∞, for any nontrivial torsion-free L_{∞}-structure N modelling T_{free,∞}.

Proof: Again the proof of this lemma is analogous to the proof of Lemma 6.8. Let t_{k}(x), s_{ℓ}(x) be L_{∞}-terms such that [ϕ(x) ∧ ¬ψ(x)] ↔ ⋁_{ℓ}[⋀_{k} t_{k}(x) = 0 ∧ s_{ℓ}(x) ≠ 0]. Each L_{∞}-term t_{k} (respectively s_{ℓ}) is equivalent to a term of the form ∑īλb̄ ; ī(x) · rī (respectively ∑j̄λb̄ ; j̄(x) · sj̄), where rī, sj̄ ∈ R − {0} (see Lemma 4.1). Notice that we may assume that the λ functions have the same superscript b̄ ∈ B^{n}. Also, for the clarity of the exposition, we reverted to the tuple notation for their subscripts which might not have the same length. We prove by induction on the maximal length r of ī for functions λī occuring in ϕ and ψ that, as in 6.8, a system Σ of the form ⋀_{k} t_{k}(x) = 0 ∧ s_{ℓ}(x) ≠ 0 has infinitely many “unequivalent” solutions if it has one.

In other words there is a quantifier-free L_{R}-formula Θ such that Σ is equivalent to Θ(λb̄ ; ī(x) : i of length ≤ r). We know the result for r = 0 as Lemma 6.7 also holds for infinite e. Now, for e > 0, Σ is equivalent to

[Θ(λb̄ ; ī(x) : i of length ≤ r) ∧ x = ∑ λb̄ ; ī(x) · t · m_{n,ī}(b̄)]

∨Θ((λb̄ ; ī(x) : i of length < r), ¯0).

Using induction hypothesis, we have only to deal with the first term of the disjunction, and, by 6.7 again, if such a system has in some non zero torsion-free model N of T_{free,∞} a solution x having the (b̄, r)-t.p., then it has one in any such structure. Therefore, we may proceed as in the finite imperfection degree case to produce unequivalent solutions. □

Proposition 7.4. T_{∞} admits quantifier elimination and is complete; for any countable B_{0} ⊂ B, the corresponding reduct is decidable.

Proof: As in Corollary 6.12, we take a recursive presentation of the ring 𝔽_{p}(B_{0})[t; α]. □

References

[1] Blossier, T., Sous-groupes infiniment définissables du groupe additif d’un corps séparablement clos in “Ensembles minimaux localement modulaires”, thèse de doctorat, Université Paris 7, 20001.

[2] Cohn, P. M., Skew fields, Encyclopedia of mathematics and its applications, edited by G.-C. Rota, volume 57, Cambridge University Press, 1995.

[3] Delon, F., Idéaux et types sur les corps séparablement clos, Supplément au Bull. de la S.M.F., Mémoire 33, Tome 116, 1988.

[4] Dellunde, P., Delon F., Point F., The theory of modules of separably closed fields-2, preprint.

[5] Ershov Y., Fields with a solvable theory, Doklady 174, pp. 19-20, 1967; English translation in Soviet Mathematics 8, pp. 575-576, 1967.

[6] Jacobson N., Basic Algebra I, second edition, Freeman, 1985.

[7] Martin G.A., Definability in reducts of algebraically closed fields, J.Symb.Logic 53, pp. 188-199, 1988.

[8] Ore O., On a special class of polynomials, Trans. Amer. Math. Soc. 35, pp. 559-584, 1933.

[9] Ore O., Theory of non-commutative polynomials, Ann. Math. 34, pp. 480-508, 1933.

[10] Prest M., Model theory and modules, L.M.Soc. Lecture Notes Series 130, Cambridge University Press, 1988.

[11] Wood C., Notes on the stability of separably closed fields, J.Symb.Logic 44, pp. 412-416, 1979.

[12] Ziegler M., Model Theory of modules, Annals of Pure and Applied Logic 26, pp. 149-213, 1984.

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.P.R.E.S.A.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