Almost Gorenstein rings

Abstract

The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra \m : \m of \m is a Gorenstein ring is solved in full generality, where \m denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored