广义FP—内射模、广义平坦模与某些环

左（右）R－模A称为GFP－内射模，如果ExtR(M,A)=0对任－2－表现R－模M成立；左（右）R－模称为G－平坦的，如果Tor1^R(M,A)=0(Tor1^R(AM)=0)对于任一2－表现右（左）R－模M成立；环R称左（右）R－半遗传环，如果投射左（右）R－模的有限表现子模是投射的，环R称为左（右）G－正而环，如果自由左（右）R－模的有限表现子模为其直和项，研究了GFP－内射模和G－平坦模的一些性质，给出了它们的一些等价刻划，并利用它们刻划了凝聚环，G－半遗传环和G－正则环。     

A Generalization of Auslander's Last Theorem

Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen–Macaulay approximation. Yoshimo extended Auslander's result to local Cohen–Macaulay rings admitting a dualizing module.Over a local Gorenstein ring the finitely generated maximal Cohen–Macaulay modules are the finitely generated Gorenstein projective modules so in fact Auslander's theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslander's theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslander's sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslander's theorem when R is Gorenstein.     

A-endomorphismes de A{X}

The idea to write this paper was the completion of the paper of N. Sankaran (cf.[6]). In fact, in his theorem, he suppose that all automorphisms of the restricted power series rings with respect to the m-adic topology of the maximal ideal of a local ring is anautomorphism by substitution. But this is true for all injective endomorphisms (see our Proposition 1.5.). We study here different classes of endomorphisms of the restricted power series rings with respect to an arbitrary non nilpotent ideal in a commutative ring. To do this, we follow very closely the techniques of M.J. O'Malley concerning the rings of power series, but adapted to the case of the rings of restricted power series (cf.[3],[4]).     

Modules Whose Endomorphism Rings are Division Rings

The well-known Schur's Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End _R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic.     

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. We provide some preliminaries on quasi-projective modules over commutative rings. Then we investigate the correlation with the well-known Prüfer conditions; that is, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky’s theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz’s related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. We then examine various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prüfer rings.     

On the Krull―Schmidt Theorem for Artinian Modules

A property of rings generalizing commutativity is introduced. If a ring satisfies this property, then the Krull--Schmidt theorem holds for Artinian modules over the ring. In particular, this property is fulfilled for local rings of finite rank and for rings such that their centers are surjectively mapped by the natural projection onto the factor with respect to the radical of the ring. A local ring for which the property fails is constructed; for the direct decompositions of Artinian modules over this ring there appear anomalies similar to the anomalies of direct decompositions of torsion-free Abelian groups of finite rank. Bibliography: 6 titles.     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

Positivity of intersection multiplicity over a two-dimensional base

We prove the positivity of Serre's Intersection Multiplicity for regular local rings that are essentially smooth over a two-dimensional, regular base. Afterward, we apply this result to prove a transversality theorem for unramified regular local rings via a local analysis on the blowup.     

BOUNDS ON MULTIPLICITIES OF LOCAL RINGS

ABSTRACT

We consider an inequality of C. Lech on the Samuel multiplicities of ideals primary to the maximal ideal of a local ring. We give an analogue of the inequality for Hilbert-Kunz multiplicities in prime characteristic, as well as some improvements on Lech's result for Samuel multiplicities of rings of equal characteristic.     

An extension of Iwasawa's theorem on finitely generated modules over power series rings

This paper describes a structure theorem for finitely generated modules over power series rings O[[T]], where O is a maximal order in a semisimple Q_p-algebra of finite dimension over Q_p, extending Iwasawa's structure theorem (the case O=?p). A particular case of such power series ring is the ring Λ[Δ], where Λ is the power series ring ?_p?T? and Δ is a finite group of order prime to p. Several applications are given, including a new proof of a result of Iwasawa important for the relationship between Hecke characters and certain Galois representations for CM fields.     

模范畴Recollement的Koenig定理

Koenig定理描述了环的导出范畴允许recollement的一个充分必要条件.本文给出环的模范畴版本的Koenig定理及其应用.应用一是可以导出Morita等价定理,应用二是可以描述三角矩阵环与模范畴的recollement之间的密切联系.     

On Almost Valuation and Almost Bézout Rings

In this paper we extend the notion of almost valuation and almost Bézout domains to arbitrary commutative rings, and we investigate the transfer of these properties to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide new classes of commutative rings satisfying these properties. As an immediate consequence, we show the failure of Anderson–Zafrullah's theorem on the integral closure of an almost valuation domain beyond the context of integral domains.     

The Krull―Schmidt Theorem for Henselian Rings

A necessary and sufficient condition on a local ring over which all indecomposable finite-dimensional algebras are local is found. The KrullSchmidt theorem for a class of rings that includes both the Henselian valuation rings and the rings of integers of multidimensional fields is proved. Bibliography: 2 titles.     

Abelianization for hyperkähler quotients

We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkähler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to quiver varieties, and compute as an example the ordinary and equivariant cohomology rings of a hyperpolygon space.     

F-rationality of certain Rees algebras and counterexamples to “Boutot's Theorem” for F-rational rings

F-rational rings are defined for rings of characteristic p > 0 using the Frobenius endomorphism and corresponds to rational singularities in characteristic 0. We study F-rationality of certain Rees algebras and prove that every Cohen-Macaulay local ring with isolated singularity and negative a-invariant has a Rees algebra which is F-rational. As a consequence, we find that “Boutot's Theorem” asserting that a pure subring of a rational singularity is a rational singularity is not true for a F-rational ring.     

Representation theory of MV-algebras

In this paper we develop a general representation theory for MV-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of MV-algebras and MV-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any MV-algebra is isomorphic to the MV-algebra of all global sections of a sheaf of MV-chains on a compact topological space. This result is intimately related to McNaughton’s theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton’s theorem. In spite of the language used in this abstract, we have written this paper in the hope that it can be read by experts in MV-algebras but not in sheaf theory, and conversely.     

Counter-examples to non-noetherian Elkik's approximation theorem

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Ascent of properties for absolutely flat homomorphisms

We show that if A – B is an absolutely flat homomorphism of consultative rings and A is arithmetical, then B is such, thus generalising a previous result (see [9]) on valutation rings.

To prove the above theorem we show first that it is true when A is a local ring and B is a local ind-étale homomorphism (in par ticular if B is a strict henselization of A ) , and we apply the following general fact:a local property which ascends to strict henselization and descends by faithful flatness ascends also by ab solutely flat homomorphlsms. This last result also applies to other properties, such as locally noetherian, geometrically unibranche, R_n,S_n, Cohen-Macaulay.     

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.