One-dimensional almost Gorenstein rings. (English) Zbl 0874.13018

Barucci, Valentina; Fröberg, Ralf

J. Algebra 188, No. 2, 418-442 (1997).

Let \(A\) be an one-dimensional local Cohen-Macaulay ring with finite closure \(\bar A\). It is well-known that \(\ell (\bar A/A) \geq \ell (A/C) + \text{type}(A)-1\), where \(C = A:\bar A\). If \(A\) is a Gorenstein ring, then \(\text{type}(A) = 1\) and \(\ell(\bar A/A) = \ell(A/C)\). For this reason the authors call \(A\) an almost Gorenstein ring if \(\ell(\bar A/A) = \ell(A/C) + \text{type}(A) - 1\). They study this class of local rings under the assumption that \(A\) has a canonical ideal \(K\) such that \(A \subseteq K \subseteq \bar A\). If \(A\) is analytically irreducible and residual, one can assign with \(A\) a numerical semigroup \(v(A)\). E. Kunz [Proc. Am. Math. Soc. 25, 748-751 (1970; Zbl 0197.31401)] proved that \(A\) is Gorenstein if and only if \(v(A)\) is symmetric. Inspired of this result, the authors show that almost Gorenstein rings and some of their properties can be similarly characterized in terms of \(v(A)\).

Reviewer: Ngo Viet Trung (Hanoi)

MSC:

13H10

Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Keywords:

almost Gorenstein ring; Kunz ring; analytically irreducible ring

Citations:

Zbl 0197.31401

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