Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T18:56:56.862Z Has data issue: false hasContentIssue false
  • English
  • Français

One-Dimensional Monoid Rings with n-Generated Ideals

Published online by Cambridge University Press:  20 November 2018

James S. Okon  and
J. Paul Vicknair
James S. Okon
Affiliation:
Department of Mathematics California State University San Bernardino, California 92407 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

Keywords

20M2513E1513F99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 3 , 01 September 1993 , pp. 344 - 350
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Arnold, J. T. and Matsuda, R., The n-generator property for semigroup rings, Houston J. Math. 12(1986), 345–356.Google Scholar
2. Gilmer, R., Commutative Semigroup Rings, University of Chicago Press, Chicago, 1984.Google Scholar
3. Greither, C., On the two generator problem for ideals of a one-dimensional ring, J. Pure Appl. Algebra 24(1982), 265–276.Google Scholar
4. Hardy, B. and Shores, T., Arithmetical semigroup rings, Canad. J. Math. 32( 1980), 1361–1371.Google Scholar
5. Karpilovsky, G., Commutative Group Algebras, Marcel Dekker, New York, 1983.Google Scholar
6. Matsuda, R., Torsion-free abelian semigroup rings, V, Bull. Fac. Sci., Ibaraki Univ., Math. (A) 11(1979), 1–37.Google Scholar
7. Matsuda, R., n-Generator property of a polynomial rings, Bull. Fac. Sci., Ibaraki Univ., Math. (A) 16(1984), 17–23.Google Scholar
8. Okon, J., Rush, D. and Vicknair, J. P., Commutative semigroup rings with two-generated ideals, J. London Math. Soc, (2) 45(1992), 417–432.Google Scholar
9. Okon, J. and Vicknair, J. P., Group rings with n-generated ideals, Comm. Algebra 20(1992), 189–217.Google Scholar
10. Raynaud, M., Anneaux locaux Henseliens, Lecture Notes in Math. (169), Springer-Verlag, Berlin, 1970.Google Scholar
11. Rush, D. E., Rings with two-generated ideals, J. Pure Appl. Algebra 73(1991), 257–275.Google Scholar
12. Shalev, A., Dimension subgroups, nilpotency indices, and the number of generators of ideals in p-group algebras,]. Algebra 129(1990),412–438.Google Scholar