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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rings with finite Gorenstein injective dimension


Author: Henrik Holm
Journal: Proc. Amer. Math. Soc. 132 (2004), 1279-1283
MSC (2000): Primary 13D02, 13D05, 13D07, 13H10; Secondary 16E05, 16E10, 16E30
Published electronically: November 7, 2003
MathSciNet review: 2053331
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Abstract: In this paper we prove that for any associative ring $R$, and for any left $R$-module $M$ with finite projective dimension, the Gorenstein injective dimension $\mathrm{Gid}_R M$ equals the usual injective dimension $\mathrm{id}_R M$. In particular, if $\mathrm{Gid}_R R$ is finite, then also $\mathrm{id}_R R$ is finite, and thus $R$ is Gorenstein (provided that $R$ is commutative and Noetherian).


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Additional Information

Henrik Holm
Affiliation: Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, 2100 K\obenhavnØ, Danmark
Email: holm@math.ku.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07466-5
PII: S 0002-9939(03)07466-5
Keywords: Gorenstein dimensions, homological dimensions, Gorenstein rings
Received by editor(s): January 28, 2003
Published electronically: November 7, 2003
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2003 American Mathematical Society