On GI-Injective Modules

Abstract

Let R be a ring. A left R-module M is called GI-injective if for any Gorenstein injective left R-module N. It is shown that a left R-module M over any ring R is GI-injective if and only if M is a kernel of a Gorenstein injective precover f: A → B of a left R-module B with A injective. Suppose R is an n-Gorenstein ring, we prove that a left R-module M is GI-injective if and only if M is a direct sum of an injective left R-module and a reduced GI-injective left R-module. Then we investigate GI-injective dimensions of modules and rings. As applications, some new characterizations of the weak (Gorenstein) global dimension of coherent rings are given.

Key Words:

• Coherent ring
• GI-injective dimension
• GI-injective module
• Gorenstein injective module
• n-Gorenstein ring
• Weak (Gorenstein) global dimension

2010 Mathematics Subject Classification:

• 16E30
• 16E10
• 16E50

ACKNOWLEDGMENTS

I would like to thank the referee for many considerable inputs and suggestions, which have improved this paper. This work was partially supported by NSFC (No. 11171240), and the Scientific Research Foundation of CUIT (No. KYTZ201201).

Notes

Communicated by J. L. Gomez Pardo.