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Réponses 1-21 (sur 21)

862.16027
Li, Huishi; Van Oystaeyen, Freddy
Zariskian filtrations. (English)
[B] K-Monographs in Mathematics. 2. Dordrecht: Kluwer Academic Publishers, ix, 252 p. Dfl. 195.00; \$ 127.00; \sterling 86.00 (1996). [ISBN 0-7923-4184-8/hbk]
This book is the first to present a complete theory of filtrations on associative rings, combining techniques stemming from number theory related to valuations, with facts originating in the study of rings of differential operators on varieties.\par It is divided into four chapters, each of which is subdivided into sections. The first chapter, ``Filtered Rings and Modules'', is devoted to the general theory of filtered rings and modules together with their associated graded objects.\par In the second chapter, ``Zariskian Filtrations'', the authors define (non-commutative) Zariskian filtrations and provide several characterizations of these, which turns out that several notions appearing in the literature (earlier treated as different) are in fact equivalent to the notion of Zariskian filtration. Zariskian filtrations on simple Artinian rings are studied. Slightly extending results of Quillen, the calculation of $K\sb 0$ for a ring with Zariskian filtration are also studied.\par Since the late sixties, various Auslander regularity conditions have been widely investigated in both commutative and non-commutative cases. In particular, the dimension and multiplicity theory for modules, the pure (holonomic) module theory over noncommutative Auslander regular rings have been developed in the study of rings of differential operators and enveloping algebras. In the third chapter, ``Auslander Regular (graded) Rings'', the authors present a general study of Auslander regularity of filtered rings and graded rings and then generalize the important Gabber-Kashiwara theorem and Roos theorem to Zariski rings with commutative associated graded rings and consequently the pure module theory over Zariski rings is developed as well.\par In the fourth chapter, ``Microlocalization of Filtered Rings and Modules, Quantum Sections and Gauge Algebras'', the authors connect to a trendy topic: quantum algebras. Via algebraic microlocalization and rings of quantum sections they obtain methods for putting a ``scheme''-structure on a big class of quantum algebras. The generalized Gauge algebra is introduced. The filtered techniques, in particular the lifting results from the associated graded ring to the Rees ring, provide an iterative construction of nice gauge algebras. The approach via quantum sections fits nicely in the categorical approach a la M. Artin and leads to the definition of ``schematic algebras''; this class includes iterated Ore extensions, gauge algebras iterated from the quantum plane, colour Lie superalgebras, quantum Weyl algebras etc. Finally, the authors provide concrete calculations of rings of quantum sections for enveloping algebras of finite dimensional Lie algebras and point out how to use this in a calculation of the point variety of homogenized enveloping algebras.
[ Y.Kurata (Hiratsuka) ]
Mots clés: filtered rings; schematic algebras; filtered modules; filtrations on associative rings; rings of differential operators; Zariskian filtrations; simple Artinian rings; Auslander regularity; Auslander regular rings; enveloping algebras; graded rings; Zariski rings; quantum algebras; algebraic microlocalization; rings of quantum sections; Rees rings; gauge algebras; quantum sections; iterated Ore extensions; colour Lie superalgebras; quantum Weyl algebras

Classification:: *16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16-02 Research monographs (assoc. rings and algebras); 16W50 Graded associative rings and modules

859.16037
Li, Huishi
Some recent developments in filtered ring and graded ring theory in China. (English)
[CA] Cao, X. H. (ed.) et al., Rings, groups, and algebras. New York, NY: Marcel Dekker, Lect. Notes Pure Appl. Math. 181, 145-157 (1996). [ISBN 0-8247-9733-7/pbk]
The paper is a survey of recent results on various topics as: characterizations of rings with Zariskian filtrations, their $K\sb 0$ group, Zariski rings with Auslander-regular or Auslander-Gorenstein associated graded ring, local-global results, etc. Then some graded homological properties are discussed in connection with their ungraded correspondents.
[ S.Raianu (Bucuresti) ]
Mots clés: rings with Zariskian filtrations; $K\sb 0$ groups; Zariski rings; associated graded rings

Classification:: *16W50 Graded associative rings and modules; 16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16E30 Homological functors on associative modules

855.16012
Li, Huishi
Global dimension of graded local rings. (English)
[J] Commun. Algebra 24, No.7, 2399-2405 (1996). [ISSN 0092-7872]
The main result of this paper is the following: if $A=\bigoplus\sb{n\geq 0}A\sb n$ is a positively graded Noetherian ring, and if $A\sb 0/J(A\sb 0)$ is semisimple ($J(A\sb 0)$ is the Jacobson radical of $A\sb 0$), then the global dimension of $A$ is equal to the projective dimension of $A\sb 0/J(A\sb 0)$.
[ S.Raianu (Bucuresti) ]
Mots clés: positively graded Noetherian rings; global dimension; projective dimension

Classification:: *16E10 Homological dimensions (assoc. rings and algebras); 16W50 Graded associative rings and modules

849.16037
Li, Huishi
Lifting Ore sets of Noetherian filtered rings and applications. (English)
[J] J. Algebra 179, No.3, 686-703, Art. No.0031 (1996). [ISSN 0021-8693]
If $R$ is a filtered ring such that the Rees ring $\widetilde{R}$ of $R$ is left Noetherian, and $T$ is a left Ore set of the associated graded ring $G(R)$ consisting of homogeneous elements, then the saturated set $S=\{s\in R,\ \sigma(s)\in T\}$ is a left Ore set of $R$. The above lifting property is used to obtain local-global results on $R$.
[ S.Raianu (Bucuresti) ]
Mots clés: left Noetherian filtered rings; Rees rings; left Ore sets; graded rings; homogeneous elements

Classification:: *16W50 Graded associative rings and modules; 16U20 Rings of quotients (assoc. rings and algebras); 16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16P50 Localization and associative Noetherian rings; 16P40 Associative Noetherian rings and modules

820.16038
Li, Huishi
Note on the homological dimension of graded rings. (English)
[J] Commun. Algebra 22, No.15, 6225-6237 (1994).
If $A$ is a $G$-graded ring, and $c$ is an element of the center of the group $G$, then the ``$c$-mixed'' gradation on the skew polynomial ring in one indeterminate over $A$ is introduced. Then the relations between graded homological dimensions of $A$ and graded homological dimensions of the skew polynomial ring (with respect to the $c$-mixed grading) are discussed. Applications to strongly graded rings over the integers and invertible ideals are then given.
[ S.Raianu (Bucuresti) ]
Mots clés: $G$-graded rings; skew polynomial rings; graded homological dimensions; $c$-mixed gradings; strongly graded rings

Classification:: *16W50 Graded associative rings and modules; 16E10 Homological dimensions (assoc. rings and algebras); 16S36 Ordinary and skew polynomial rings and semigroup rings

820.16039
Li, Huishi
Noetherian gr-regular rings are regular. (English)
[J] Chin. Ann. Math., Ser. B 15, No.4, 463-468 (1994).
Let $A$ be a left noetherian ring which is graded by the group $\bbfZ$. If every finitely generated graded $A$-module has finite projective dimension, then it is shown that every finitely generated $A$-module has finite projective dimension.
[ C.Nastasescu (Bucuresti) ]
Mots clés: left Noetherian rings; finitely generated graded modules; finite projective dimension; finitely generated modules

Classification:: *16W50 Graded associative rings and modules; 16E10 Homological dimensions (assoc. rings and algebras); 16P40 Associative Noetherian rings and modules; 16E50 Von Neumann regular rings and generalizations

816.16032
Li, Huishi
On the stability of graded rings. (English)
[J] J. Algebra 169, No.1, 274-286 (1994).
If $R$ is a ring graded by the integers, which is noetherian and left graded regular, such that every finitely generated graded projective $R$-module is graded stably free, then it is shown that the corresponding ungraded property holds. Some applications to Rees rings of Zariskian filtered rings are given.
[ S.Raianu (Bucuresti) ]
Mots clés: left graded regular rings; graded projective $R$-modules; graded stably free modules; Rees rings; Zariskian filtered rings

Classification:: *16W50 Graded associative rings and modules; 16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16D40 Free, etc. modules and ideals (assoc. rings and algebras)

786.16022
Li, Huishi
Rees rings of grading filtrations and an application to Weyl algebras. (English)
[J] Commun. Algebra 21, No.8, 2967-2972 (1993).
Let $R$ be a \bbfZ-graded ring. This note studies connections between different filtrations or gradations, e.g. the grading filtrations on $R$ and the sign gradations on the polynomial ring $R[t]$. The results are applied to Weyl algebras.
[ S.Raianu (Bucuresti) ]
Mots clés: \bbfZ-graded ring; grading filtrations; sign gradations; polynomial ring; Weyl algebras

Classification:: *16W50 Graded associative rings and modules; 16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16S36 Ordinary and skew polynomial rings and semigroup rings

776.16020
Li, Huishi; Van Oystaeyen, F.
(Oystaeyen, F.Van)
Sign gradations on group ring extensions of graded rings. (English)
[J] J. Pure Appl. Algebra 85, No.3, 311-316 (1993).
If $R$ is a $G$-graded ring ($G$ is a group) and $H$ is a submonoid of the center $Z(G)$, then a $G$-gradation on the semigroup ring $RH$ is defined. Applications to graded rings and $I$-adic filtrations are given.
[ S.Raianu (Bucuresti) ]
Mots clés: submonoid of center; semigroup ring; graded rings; $I$-adic filtrations

Classification:: *16W50 Graded associative rings and modules; 16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16S36 Ordinary and skew polynomial rings and semigroup rings; 16S34 Group rings (assoc. rings); 20M25 Semigroup rings, multiplicative semigroups of rings

772.16016
Li, Huishi; Van Oystaeyen, Freddy
(Oystaeyen, F.van)
Dehomogenization of gradings to Zariskian filtrations and applications to invertible ideals. (English)
[J] Proc. Am. Math. Soc. 115, No.1, 1-11 (1992).
The authors extend the method of dehomogenizing graded rings to noncommutative graded rings, dehomogenizing suitably graded rings to Zariski filtered rings and deriving, in a very elementary way, homological properties related to Auslander regularity and the Gorenstein property for noncommutative rings. As an application they study the lifting of such properties from a quotient modulo an invertible ideal.
[ Xue Weimin (Fouzhou) ]
Mots clés: dehomogenizing graded rings; Zariski filtered rings; Auslander regularity; Gorenstein property; lifting

Classification:: *16W50 Graded associative rings and modules; 16D40 Free, etc. modules and ideals (assoc. rings and algebras); 16W60 Filtrations and valuations, etc. (assoc. rings and algebras)

758.16014
Li, Huishi
(Li, H.S.)
Note on microlocalizations of filtered rings and the embedding of rings in skewfields. (English)
[J] Bull. Soc. Math. Belg., Ser. A 43, No.1/2, 49-57 (1991).
``In this note, we discuss some more properties of microlocalizations of filtered rings, and moreover, we prove that for an arbitrary Lie algebra $L$, if $U(L)$ denotes the enveloping algebra of $L$ with the standard filtration $F(U(L))$, then the skewfield $D$ constructed by P. M. Cohn in 1961 such that $U(L)\subset D$ is just a certain type of microlocalization of $U(L)$''.
[ S.Elliger (Bochum) ]
Mots clés: microlocalizations; filtered rings; Lie algebra; enveloping algebra; skewfield

Classification:: *16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16U20 Rings of quotients (assoc. rings and algebras); 17B35 Universal enveloping algebras (Lie algebras); 16K40 Infinite dimensional and general division rings

753.16003
Li, Huishi; Van Oystaeyen, F.
(Oystaeyen, F.van)
Global dimension and Auslander regularity of Rees rings. (English)
[J] Bull. Soc. Math. Belg., Ser. A 43, No.1/2, 59-87 (1991).
A ring $R$ with a filtration $\{F\sb nR\}$ is a Zariski ring if $F\sb{-1}R\subset J(F\sb 0R)$. If $M$ is an $R$-module the grade of $M$, $j(M)$, is defined by $j(M):=\inf\{k\mid \text{Ext}\sp k(M,R)\ne0\}$ and a noetherian ring $R$ with finite global dimension $\mu$ is said to be regular if every finitely generated $R$-module satisfies the Auslander condition: for each finitely generated $R$-module $M$ and each $0\le k\le \mu$, if $0\ne N$ is a submodule of $\text{Ext}\sp k(M,R)$ then $j(N)\le k$. Various results concerning homological dimension and regularity are proved in this paper, for example: Let $R$ be a Zariski ring and suppose that the associated graded ring $G(R)$ is regular noetherian then the associated Rees ring $\tilde R$ of $R$ is regular noetherian and $\text{gl.}\dim(\tilde R)=1+\text{gl.}\dim G(R)$.
[ T.H.Lenagan (Edinburgh) ]
Mots clés: filtration; Zariski ring; noetherian ring; global dimension; homological dimension; graded ring; regular noetherian; Rees ring; Auslander condition

Classification:: *16E10 Homological dimensions (assoc. rings and algebras); 16P40 Associative Noetherian rings and modules; 16W50 Graded associative rings and modules

744.16024
Li, Huishi
Note on pure module theory over Zariskian filtered ring and the generalized Roos theorem. (English)
[J] Commun. Algebra 19, No.3, 843-862 (1991).
This paper is concerned with homological properties of modules over a Zariskian filtered ring $R$ whose associated graded ring, $G(R)$, is a regular Noetherian ring; i.e. $G(R)$ is a Noetherian ring with finite global dimension and every finitely generated module satisfies the Auslander condition. In this situation, if the grade of $M$ is defined by $j(M):=\inf\{k\mid\hbox{Ext}\sp k(M,R)\neq 0\}$ then it is shown that $j(M)=j(G(M))=j(\tilde M)=j(\hat M)$, where $\tilde M$ is the induced module over the Rees ring and $\hat M$ the induced module over the completion. Homological properties such as purity can then be passed to and fro between the various rings. There then follows a specialisation to the case that $G(R)$ is also commutative and here holonomic modules are characterised in terms of heights of prime ideals in their characteristic varieties.
[ T.H.Lenagan (Edinburgh) ]
Mots clés: Zariskian filtered ring; graded ring; regular Noetherian ring; global dimension; Rees ring; induced module; completion; purity; holonomic modules; heights of prime ideals; characteristic varieties

Classification:: *16W50 Graded associative rings and modules; 13E05 Noetherian rings and modules; 13A30 Associated graded rings of ideals and related topics; 16W60 Filtrations and valuations, etc. (assoc. rings and algebras)

722.16019
Li, Huishi; Van Oystaeyen, F.; Wexler-Kreindler, E.
(Oystaeyen, F.van)
Zariski rings and flatness of completion. (English)
[J] J. Algebra 138, No.2, 327-339 (1991).
The paper studies filtered rings with a separated filtration such that the associated graded ring is noetherian. Such filtrations are Zariskian exactly when the complete ring is faithfully flat as a right module. The flatness property of the completion corresponds then exactly to the comparison condition for the filtration. Several applications are given.
[ C.Nastasescu (Bucuresti) ]
Mots clés: filtered rings; separated filtration; graded ring; complete ring; faithfully flat; completion

Classification:: *16W50 Graded associative rings and modules; 16P40 Associative Noetherian rings and modules; 16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16D40 Free, etc. modules and ideals (assoc. rings and algebras)

761.16016
Li, Huishi; Van Oystaeyen, F.
(Oystaeyen, F.van)
Filtrations on simple Artinian rings. (English)
[J] J. Algebra 132, No.2, 361-376 (1990).
After some results concerning commutative filtered rings we turn to the study of filtrations on Artinian (left and right) rings with special attention for simple Artinian rings and skewfields. We rediscover discrete valuation rings of a skewfield in the sense of O. Schilling as exhaustive and separated filtrations having an associated graded ring without homogeneous zero-divisors (see Theorem 3.2). In the case of P.I. algebras we describe the filtrations that correspond to the maximal orders over discrete valuation rings (see Theorem 3.8) in terms of the Jacobson radical property of the filtration. The case where the associated graded ring of a filtered simple Artinian ring is a semiprime P.I. ring is reduced to the prime case, by using microlocalization; without P.I. hypothesis an extra assumption is necessary in order to arrive at the same conclusion (see resp. Theorem 3.9, Theorem 3.10).
Mots clés: simple Artinian rings; skewfields; discrete valuation rings; separated filtrations; associated graded ring; P.I. algebras; maximal orders; filtered simple Artinian ring; microlocalization

Classification:: *16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 16W50 Graded associative rings and modules; 16K40 Infinite dimensional and general division rings

722.16004
Li, Huishi; Van den Bergh, M.; Van Oystaeyen, F.
(Bergh, M.van den; Oystraeyen, F.van)
Global dimension and regularity of Rees rings for non-Zariskian filtrations. (English)
[J] Commun. Algebra 18, No.10, 3195-3208 (1990).
The following pretty result is proved here: Let A be a (left and right) Noetherian ring containing a central nonzero-divisor, x, such that A/xA has finite global dimension. Then $$ gl.\dim.(A)=\max \{1+gl.\dim.(A/xA),gl.\dim.(A\sb x)\}, $$ (where $A\sb x$ denotes the localization of A at the powers of x). This result is applied to obtain a relation between the global dimensions of a filtered ring R and of its associated graded ring G(R) and Rees ring $\tilde R$ (provided this last ring is Noetherian). Similar results for rings with Zariskian filtrations have been obtained by {\it H. Li} and {\it F. Van Oystaeyen} [Global dimension and Auslander regularity of Rees rings (preprint)]. There is also an application to Auslander regularity: With A and x as above, A is Auslander regular provided A/xA and $A\sb x$ are.
[ K.A.Brown (Glasgow) ]
Mots clés: Noetherian ring; finite global dimension; localization; filtered ring; graded ring; Rees ring; Auslander regularity

Classification:: *16E10 Homological dimensions (assoc. rings and algebras); 16W50 Graded associative rings and modules; 16P40 Associative Noetherian rings and modules; 16W60 Filtrations and valuations, etc. (assoc. rings and algebras); 18G20 Homological dimension

709.16023
Li, Huishi; Van den Bergh, M.; Van Oystaeyen, F.
(Bergh, M.van den; Oystaeyen, F.van)
Note on the $K\sb 0$ of rings with Zariskian filtration. (English)
[J] K-Theory 3, No.6, 603-606 (1990).
Let R be a non-commutative Zariski ring, whose associated graded ring G(R) has finite global dimension. It is shown that there is an injection $K\sb 0(R)\hookrightarrow K\sb 0(G(R))$. This result can be applied to certain rings of differential operators.
[ C.Nastasescu ]
Mots clés: Zariski ring; graded ring; finite global dimension; rings of differential operators

Classification:: *16W50 Graded associative rings and modules; 16E20 K-theory of noncommutative rings; 19A49 $K\sb 0$ of other rings; 16S32 Associative rings of differential operators; 16E70 Other assoc. rings of low global or flat dimension

691.16003
Li, Huishi; Van Oystaeyen, F.
(Oystaeyen, F.van )
Zariskian filtrations. (English)
[J] Commun. Algebra 17, No.12, 2945-2970 (1989).
The paper studies non-commutative Zariski rings. Zariski rings with noetherian Rees ring and with commutative associated graded ring are investigated.
[ C.Nastasescu ]
Mots clés: Zariski rings; noetherian Rees ring; graded ring

Classification:: *16W50 Graded associative rings and modules; 16U80 Generalizations of commutativity (assoc. rings and algebras); 13E05 Noetherian rings and modules; 13J99 Topological rings

686.16001
Li, Huishi; Van Oystaeyen, Freddy
(Oystaeyen, F.van )
Strongly filtered rings applied to Gabber's integrability theorem and modules with regular singularities. (English)
[CA] Seminaire d'algebre P. Dubreil et M.-P. Malliavin, Proc., Paris/Fr. 1987/88, Lect. Notes Math. 1404, 296-321 (1989).
[For the entire collection see Zbl. 677.00008.] \par The authors first sketch some previous work in the theory of filtered rings, in particular the technique of shifting properties from the associated graded ring on the Rees ring to the filtered ring. They further generalize the definition of involutive ideal, as used in the statement of Gabber's theorem, to filtered rings which are not necessarily E-rings, but still satisfy the original conditions on the associated graded ring. By studying properties of the microlocalization of strongly filtered rings, they are able to prove an extension of Gabber's theorem for the new notion of primitive ideal as defined above. This result is then further used to characterize finitely generated, filtered R-modules which have regular singularities along some ideal I of the associated graded ring. For holonomic modules with good filtration, it is shown to have regular singularities as an R-module may be checked by microlocalization at every graded prime of the associated graded ring. In a final section, these results are generalized to filtered rings which need not to be strongly filtered. The paper is fairly self contained, but the reader has to be well versed in graded ring theory to understand the proofs.
[ P.Nelis ]
Citations: Zbl.677.00008
Mots clés: filtered rings; associated graded ring; Rees ring; involutive ideal; Gabber's theorem; microlocalization; strongly filtered rings; primitive ideal; regular singularities; holonomic modules; good filtration; graded prime

Classification:: *16W50 Graded associative rings and modules; 16Dxx Modules, bimodules and ideals (assoc. rings and algebras)

632.16007
Li, Huishi
Semiprime modules and supernilpotent radicals. (Chinese)
[J] Acta Math. Sin. 29, 213-216 (1986).
Let M be an R-module. M is called R-semiprime if MR$\ne 0$ and the quotient ring $R/M\sb r$ is semiprime, where $M\sb r$ is the right annihilator of M in R. By using the concept of semiprime modules, a weak special class of R-modules is introduced. The author shows that if S is an Amitsur-Kurosh radical property, then S is super nilpotent if and only if there exists a weak special class of R-modules M such that the S-radical S(R) and the M-radical M(R) are equal for every ring R. The author claims that this result answers a problem posed by {\it F. A. Szasz} [in his book ``Radicals of rings'' (1981; Zbl. 461.16009), p. 91, Problem 16].
[ K.-P.Shum ]
Citations: Zbl.461.16009
Mots clés: quotient ring; right annihilator; semiprime modules; weak special class; Amitsur-Kurosh radical property; super nilpotent

Classification:: *16Nxx Radicals and radical properties of assoc. rings; 16N40 Nil and nilpotent radicals, sets, ideals, rings; 16N60 Prime and semiprime assoc. rings

592.16005
Zhou, Yiqiang; Li, Huishi
On the module-theoretic characterization of supernilpotent radical class of rings. (English)
[J] Kexue Tongbao, Foreign Lang. Ed. 30, 1303-1306 (1985).
Module-theoretic characterizations of a general radical class and a special radical class are known [see {\it F. A. Szasz}, Radicals of rings (1981; Zbl. 461.16009)]. The authors, by using two methods in the same way, characterize supernilpotent radical classes defined, in the first by the semiprime module property, a weakly special class of modules and, in the other, a class of modules of another kind.
[ G.Tzintzis ]
Citations: Zbl.461.16009
Mots clés: supernilpotent radical classes; weakly special class of modules

Classification:: *16Nxx Radicals and radical properties of assoc. rings; 16D80 Other classes of modules and ideals (assoc. rings and algebras); 16N40 Nil and nilpotent radicals, sets, ideals, rings

Réponses 1-21 (sur 21)

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