Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

Cohomology theories based on Gorenstein injective modules

In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.

     

Selforthogonal modules with finite injective dimension

The category consisting of finitely generated modules which are left orthogonal with a cotilting bimodule is shown to be functorially finite. The notion of left orthogonal dimension is introduced, and then a necessary and sufficient condition of selforthogonal modules having finite injective dimension and a characterization of cotilting modules are given.     

On finitely injective modules and locally pure-injective modules over Prüfer domains

Over Matlis valuation domains there exist finitely injective modules which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Prüfer domains which are either almost maximal, or -local Matlis, finitely injective torsion modules and complete torsion-free locally pure-injective modules correspond to each other under the Matlis equivalence. Almost maximal Prüfer domains are characterized by the property that every torsion-free complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind.

     

Gorenstein injective complexes of modules over Noetherian rings

A complex C is called Gorenstein injective if there exists an exact sequence of complexes $?\to I_{\mathrm{?1}}\to I_0\to I_1\to ?$ such that each $I_i$ is injective, $C=\mathrm{Ker}(I_0\to I_1)$ and the sequence remains exact when $\mathrm{Hom}(E,?)$ is applied to it for any injective complex E. We show that over a left Noetherian ring R, a complex C of left R-modules is Gorenstein injective if and only if $C^m$ is Gorenstein injective in R-Mod for all $m\in \mathbb{Z}$. Also Gorenstein injective dimensions of complexes are considered.     

一类广义遗传环

称环R为左亚遗传环，如果内射左R－模的商模是FG－内射的，给出了左亚遗传环的一些刻划，给出了左亚遗传环的半单环的条件，并研究了左亚遗传环的一些性质。     

Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.     

RRF rings which are not LRF

Define a ringA to be RRF (respectively LRF) if every right (respectively left)A-module is residually finite. We determine the necessary and sufficient conditions for a formal triangular matrix ring to be RRF (respectively LRF). Using this we give examples of RRF rings which are not LRF.     

On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

On finitely embedded rings

A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case.     

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

Classifying annihilating-ideal graphs of commutative artinian rings

In this article we investigate the annihilating-ideal graph of a commutative ring, introduced by Behboodi and Rakeei in [10 Behboodi, M., Rakeei, Z. (2011). The annihilating-ideal graph of commutative rings I. J. Algebra Appl. 10(4):727–739.[Crossref], [Web of Science ®] , [Google Scholar]]. Our main goal is to determine which algebraic properties of a ring are reflected in its annihilating-ideal graph. We prove that, for artinian rings, the annihilating-ideal graph can be used to determine whether the ring in question is a PIR or, more generally, if it is a dual ring. Moreover, with one trivial exception, the annihilating-ideal graph can distinguish between PIRs with different ideal lattices. In addition, we explore new techniques for classifying small annihilating-ideal graphs. Consequently, we completely determine the graphs with six or fewer vertices which can be realized as the annihilating-ideal graph of a commutative ring.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.