Gorenstein injective envelopes of Artinian modules

TL;DR

This paper proves that Artinian modules over certain rings have special Gorenstein injective envelopes, extending known results and providing new structural insights into module theory.

Contribution

It establishes the existence of special Gorenstein injective envelopes for Artinian modules with finite Gorenstein injective dimension, especially over Gorenstein rings.

Findings

Artinian modules with finite Gorenstein injective dimension have special Gorenstein injective envelopes.

Over Gorenstein rings, all Artinian modules possess such envelopes.

Envelopes are both special and Artinian.

Abstract

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.