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

##### MSC:
 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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