Identification Gorenstein rings via special semidualizing modules

‎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=۱}{\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=۰ ‎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.

کلیدواژه ها:

The Auslander-Reiten conjecture ، Semidualizing modules ، Free modules