STAR OPERATIONS ON KUNZ DOMAINS

Year 2019, Volume: 25 Issue: 25, 171 - 185, 08.01.2019

Dario Spirito

https://doi.org/10.24330/ieja.504142

Abstract

We study star operations on Kunz domains, a class of analyti-

cally irreducible, residually rational domains associated to pseudo-symmetric

numerical semigroups, and we use them to refute a conjecture of Houston, Mi-

mouni and Park. We also nd an estimate for the number of star operations

in a particular case, and a precise counting in a sub-case.

Keywords

Star operation, pseudo-symmetric semigroup, Kunz domain, star regular domain

References

• V. Barucci, D. E. Dobbs and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Amer. Math. Soc., 125(598) (1997), x+78 pp.
• H. Bass, On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8-28.
• E. G. Houston, A. Mimouni and M. H. Park, Integral domains which admit at most two star operations, Comm. Algebra, 39(5) (2011), 1907-1921.
• E. G. Houston, A. Mimouni and M. H. Park, Noetherian domains which admit only nitely many star operations, J. Algebra, 366 (2012), 78-93.
• E. G. Houston, A. Mimouni and M. H. Park, Integrally closed domains with only nitely many star operations, Comm. Algebra, 42(12) (2014), 5264-5286.
• E. Houston, A. Mimouni and M. H. Park, Star operations on overrings of Noetherian domains, J. Pure Appl. Algebra, 220(2) (2016), 810-821.
• E. Houston, A. Mimouni and M. H. Park, Star operations on overrings of Prufer domains, Comm. Algebra, 45(8) (2017), 3297-3309.
• E. G. Houston and M. H. Park, A characterization of local Noetherian domains which admit only nitely many star operations: The innite residue eld case, J. Algebra, 407 (2014), 105-134.
• J. Jager, Langenberechnung und kanonische Ideale in eindimensionalen Ringen, Arch. Math. (Basel), 29(5) (1977), 504-512.
• E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc., 25 (1970), 748-751.
• E. Kunz, Beispiel: Die kanonische Idealklasse eines eindimensionalen Cohen- Macaulay-Rings, 103, Lecture Notes in Math., Vol. 238, Springer, Berlin, (1971), 17-24.
• T. Matsuoka, On the degree of singularity of one-dimensional analytically irreducible Noetherian local rings, J. Math. Kyoto Univ., 11 (1971), 485-494.
• J. C. Rosales and P. A. Garca-Sanchez, Numerical Semigroups, Developments in Mathematics, 20, Springer, New York, 2009.
• D. Spirito, Star operations on numerical semigroups, Comm. Algebra, 43(7) (2015), 2943-2963.
• B. White, Star Operations and Numerical Semigroup Rings, Ph.D. thesis, The University of New Mexico, 2014.