Identification Gorenstein rings via special semidualizing modules

Document Type : Research Paper

Authors

1 Department of ‎Mathematics,‎‎ Imam Khomeini ‎‏‎I‎nternational University, Qazvin, Iran

2 Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

Abstract

‎Let ‎$‎(R, {\frak m})‎$ ‎be a‎ ‎Noetherian ‎local ‎ring ‎and ‎‎$‎M‎$ ‎be a ‎finitely generated ‎$‎R‎$‎-module such that ‎$‎‎{\rm Hom}_R(M,R) \cong \underset{i=1}{\overset{n}{\oplus}} C$ ‎for ‎some ‎positive ‎integer ‎‎$‎n‎$‎. We try to present new characterizations of Gorenstein rings via ‎$‎M‎$ ‎and ‎‎$‎C‎$‎. It is proved that if ‎$‎‎{\rm depth}\, R=0$ ‎and ‎‎$‎‎{\rm id}_R (M) < ‎\infty‎$ ‎then ‎‎$‎R‎$ ‎is ‎Gorenstein. Also, it is shown that‏ if ‎‎$‎M‎$ ‎is a‎ ‎Cohen-Macaulay ‎‎$‎R‎$‎-module with finite injective dimension, then ‎$‎R‎$ ‎is ‎Gorenstein.

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References

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