zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] Burch, L., On ideals of finite homological dimension in local rings, Proc. Cambridge philos. soc., 64, 941-948, (1968) · Zbl 0172.32302
[2] Corso, A.; Polini, C., Links of prime ideals and their Rees algebras, J. algebra, 178, 224-238, (1995) · Zbl 0848.13015
[3] Corso, A.; Polini, C., Reduction number of links of irreducible varieties, J. pure appl. algebra, 121, 29-43, (1997) · Zbl 0891.13008
[4] Goto, S.; Hayasaka, F., Finite homological dimension primes associated to integrally closed ideals, Proc. amer. math. soc., 130, 3159-3164, (2002) · Zbl 0995.13009
[5] Goto, S.; Matsuoka, N.; Takahashi, R., Quasi-socle ideals in a Gorenstein local ring, J. pure appl. algebra, 212, 969-980, (2008) · Zbl 1137.13014
[6] Goto, S.; Sakurai, H., The equality \(I^2 = Q I\) in Buchsbaum rings, Rend. sem. mat. univ. Padova, 110, 25-56, (2003) · Zbl 1167.13302
[7] Goto, S.; Sakurai, H., The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two, J. math. soc. Japan, 56, 1157-1168, (2004) · Zbl 1102.13003
[8] Goto, S.; Sakurai, H., When does the equality \(I^2 = Q I\) hold true in Buchsbaum rings?, (), 115-139 · Zbl 1098.13029
[9] Herzog, J.; Kunz, E., Die wertehalbgruppe eines lokalen rings der dimension 1, S.-B. heidelberger akad. wiss. math. natur. kl., 27-67, (1971) · Zbl 0212.06102
[10] Northcott, D.G.; Rees, D., Reductions of ideals in local rings, Proc. Cambridge philos. soc., 50, 145-158, (1954) · Zbl 0057.02601
[11] Polini, C.; Ulrich, B., Linkage and reduction numbers, Math. ann., 310, 631-651, (1998) · Zbl 0919.13013
[12] Wang, H.-J., Links of symbolic powers of prime ideals, Math. Z., 256, 749-756, (2007) · Zbl 1123.13018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.