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Totally Acyclic Complexes, Sergio Estrada, Xianhui Fu, Alina Iacob
Totally Acyclic Complexes, Sergio Estrada, Xianhui Fu, Alina Iacob
Department of Mathematical Sciences Faculty Publications
It is known that over an Iwanaga–Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga–Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga–Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. …
Gorenstein Injective Envelopes And Covers Over Two Sided Noetherian Rings, Alina Iacob
Gorenstein Injective Envelopes And Covers Over Two Sided Noetherian Rings, Alina Iacob
Department of Mathematical Sciences Faculty Publications
We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.