Gorenstein Injective Envelopes of Artinian Modules

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.

Key Words:

• Gorenstein injective dimension
• Gorenstein injective envelopes
• Gorenstein injective modules
• Gorenstein projective covers
• Gorenstein projective modules
• Injective covers
• Henselian rings

2010 Mathematics Subject Classification:

• 13D05
• 13E10

ACKNOWLEDGMENT

The authors wish to thank Professor Edgar E. Enochs for his interest and comments on this work.

Notes

Communicated by V. Walther.