Gorenstein injective envelopes and covers over two sided noetherian rings

TL;DR

This paper establishes that Gorenstein injective modules form an enveloping and covering class over certain noetherian rings, and explores their relationship with strongly cotorsion modules, especially in Gorenstein rings.

Contribution

It proves the enveloping and covering properties of Gorenstein injective modules over two-sided noetherian rings with specific conditions, and characterizes when these modules coincide with strongly cotorsion modules.

Findings

Gorenstein injective modules are enveloping and covering over specified rings.

Character modules of Gorenstein injective modules are Gorenstein flat.

Gorenstein injective modules coincide with strongly cotorsion modules iff the ring is Gorenstein.

Abstract

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.