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Quasi-socle ideals in Gorenstein numerical semigroup rings. (English) Zbl 1151.13005
The authors aim to study the Polini-Ulrich conjecture [C. Polini, B. Ulrich, Math. Ann. 310, 631–651 (1998; Zbl 0919.13013)] in the one-dimensional case. The main result and its proof of this paper are given in Section 2. This result gives a generalization of the paper [S.Goto, N. Matsuoka, R. Takahashi, J. Pure Appl. Algebra 212, 969–980 (2008; Zbl 1137.13014)] in the case where the base rings are numerical semigroups rings. As an application of the main result, the authors explore in Section 3 numerical semigroups rings \(A=k[t^a, t^{a+1}] (a>1)\) over fields \(k\), where \(t\) is an indeterminate. They give a criterion for the ideal \(I=(t^s)\) : \(m^q\) to be integral over the parameter ideal \((t^s)\) in \(A\). The problem of when the ring \(G(I)\) is Cohen-Macaulay is answered in certain special cases. In Section 4, they give two examples to confirm their result.

MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
20M14 Commutative semigroups
20M25 Semigroup rings, multiplicative semigroups of rings
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