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 | References | Similar Articles | Additional Information

Abstract: Let be a commutative Noetherian local ring. We show that is Gorenstein if and only if every finitely generated -module can be embedded in a finitely generated -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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Additional Information

**Ryo Takahashi**

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

Email:
takahasi@math.shinshu-u.ac.jp

**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

Email:
yassemi@ipm.ir

**Yuji Yoshino**

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

Email:
yoshino@math.okayama-u.ac.jp

DOI:
http://dx.doi.org/10.1090/S0002-9939-10-10323-2

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.