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Proceedings of the American Mathematical Society

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On the existence of embeddings into modules of finite homological dimensions

Authors: Ryo Takahashi, Siamak Yassemi and Yuji Yoshino
Journal: Proc. Amer. Math. Soc. 138 (2010), 2265-2268
MSC (2010): Primary 13D05, 13H10
Published electronically: February 23, 2010
MathSciNet review: 2607854
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Abstract: Let $ R$ be a commutative Noetherian local ring. We show that $ R$ is Gorenstein if and only if every finitely generated $ R$-module can be embedded in a finitely generated $ R$-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.

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Ryo Takahashi
Affiliation: Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan

Siamak Yassemi
Affiliation: Department of Mathematics, University of Tehran, P. O. Box 13145-448, Tehran, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

Yuji Yoshino
Affiliation: Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan

Keywords: Gorenstein ring, Cohen-Macaulay ring, projective dimension, injective dimension, (semi)dualizing module
Received by editor(s): November 26, 2008
Received by editor(s) in revised form: May 24, 2009
Published electronically: February 23, 2010
Additional Notes: The first and second authors were supported in part by Grant-in-Aid for Young Scientists (B) 19740008 from JSPS and by grant No. 88013211 from IPM, respectively
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.