The notion of almost Gorenstein ring given by Barucci and Fröberg [2] 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.

No abstract

Abstract. Let (R, m) be a two-dimensional regular local ring with infinite residue class field. Then the Rees algebra R(I) = n≥0 I n of I is an almost Gorenstein graded ring in the sense of [6] for every m-primary integrally closed ideal I in R. IntroductionThe main purpose of this paper is to prove the following. Theorem 1.1. Let (R, m) be a two-dimensional regular local ring with infinite residue class field and I an m-primary integrally closed ideal in R. Then the Rees algebra R(I) = n≥0 I n of I is an almost Gorenstein graded ring.As a direct consequence one has the following.Corollary 1.2. Let (R, m) be a two-dimensional regular local ring with infinite residue class field. Then R(m ℓ ) is an almost Gorenstein graded ring for every integer ℓ > 0.The proof of Theorem 1.1 depends on a result of J. Verma [19] which guarantees the existence of joint reductions with joint reduction number zero. Therefore our method of proof works also for two-dimensional rational singularities, which we shall discuss in the forthcoming paper [7]. Here, before entering details, let us recall the notion of almost Gorenstein graded/local ring as well as some historical notes about it.Almost Gorenstein rings are a new class of Cohen-Macaulay rings, which are not necessarily Gorenstein, but still good, possibly next to the Gorenstein rings. The notion of these local rings dates back to the paper

No abstract

This paper explores the structure of quasi-socle ideals I = Q : m 2 in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problems as to when Q is a reduction of I and when the associated graded ring G(I ) = n≥0 I n /I n+1 is Cohen-Macaulay. Wild examples are explored.