For a left and right Noetherian ring $R$, we give some equivalent characterizations for $\_RR$ satisfying the Auslander condition in terms of flat (resp. injective) dimensions minimal injective coresolution resolution) $R$-modules. Furthermore, prove that an artin algebra $R$ condition, is Gorenstein if only subcategory consisting finitely generated modules contravariantly finite. As applicatio...

It is proved that every commutative ring whose RD-injective modules are Σ-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian (respectively simple) module is RD-injective, is given. These results can be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively th...

We give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of almost injective (or CS) modules is again almost injective (or CS) in some special cases..

Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, sec...

A new characteristic property of von Neumann regular rings is proposed in terms of annihilators of elements. An ELT fully idempotent ring is a regular ring whose simple left (or right) modules are either injective or projective. Artinian rings are characterized in terms of Noetherian rings. Strongly regular rings and rings whose two-sided ideals are generated by central idempotents are characte...

This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view toward providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved r...

Let R be a ring, a right ideal I of R is called small if for every proper right ideal K of R, I +K = R. A ring R is called right small injective if every homomorphism from a small right ideal to R R can be extended to an R-homomorphism from R R to R R. Properties of small injective rings are explored and several new characterizations are given for QF rings and P F rings, respectively.