## Tsukuba Journal of Mathematics

### Gorenstein injective modules and Ext.

#### Abstract

The aim of this paper is to characterize $n$-Gorenstein rings in terms of Gorenstein injective modules and the Ext functor. We will show that if $R$ is a left and right noetherian ring and $n$ is a positive integer, then $R$ is $n$-Gorenstein if and only if $M$ being Gorenstein injective means that Ext$^{1}$ $(L, M )=0$ for all countably generated $R$-modules $L$ of projective dimension at most $n$. In particular, if $R$ is $n$-Gorenstein, then an $R$-module $M$ is Gorenstein injective if and only if it is $U$-Gorenstein injective whenever $U$ is a free $R$-module with a countable base.

#### Article information

Source
Tsukuba J. Math., Volume 28, Number 2 (2004), 303-309.

Dates
First available in Project Euclid: 30 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1496164803

Digital Object Identifier
doi:10.21099/tkbjm/1496164803

Mathematical Reviews number (MathSciNet)
MR2105939

Zentralblatt MATH identifier
1089.16012

#### Citation

Enochs, Edgar E.; Jenda, Overtoun M. G. Gorenstein injective modules and Ext. Tsukuba J. Math. 28 (2004), no. 2, 303--309. doi:10.21099/tkbjm/1496164803. https://projecteuclid.org/euclid.tkbjm/1496164803