Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-06T15:31:11.028Z Has data issue: false hasContentIssue false
  • English
  • Français

Prime and maximal ideals in polynomial rings

Published online by Cambridge University Press:  18 May 2009

Miguel Ferrero
Miguel Ferrero
Affiliation:
Instituto de Matemática, Universidade Federal do Rio Grande do Sul, 91509-900-Porto Alegre, RS, Brazil
Rights & Permissions [Opens in a new window]

Extract

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.

In this paper we study prime and maximal ideals in a polynomial ring R[X], where R is a ring with identity element. It is well-known that to study many questions we may assume Ris prime and consider just R-disjoint ideals. We give a characterizaton for an R-disjoint ideal to be prime. We study conditions under which there exists an R-disjoint ideal which is a maximal ideal and when this is the case how to determine all such maximal ideals. Finally, we prove a theorem giving several equivalent conditions for a maximal ideal to be generated by polynomials of minimal degree.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 3 , September 1995 , pp. 351 - 362
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

REFERENCES

1.Bergman, G. M., Radicals, tensor products, and algebraicity, Israel Math. Conf. Proc. 1 (1989), 150–192.Google Scholar
2.Cisneros, E., Ferrero, M. and González, M. I., Prime ideals of skew polynomial rings and skew Laurent polynomial rings, Math. J. Okayama Univ. 32 (1990), 61–72.Google Scholar
3.Ferrero, M., Prime and principal closed ideals in polynomial rings, J. Algebra 134 (1990), 45–59.CrossRefGoogle Scholar
4.Ferrero, M., Closed and prime ideals in free centred extensions, J. Algebra 148 (1992), 1–16.CrossRefGoogle Scholar
5.Ferrero, M. and Matczuk, J., Prime ideals in skew polynomial rings of derivation type, Comm. Algebra 18, (1990), 689–710.CrossRefGoogle Scholar
6.Ferrero, M. and Parmenter, M., A note on Jacobson rings and polynomial rings, Proc. Amer. Math. Soc. 105 (1989), 281–286.CrossRefGoogle Scholar
7.Gilmer, R. W. Jr, The pseudo-radical of a commutative ring, Pacific J. Math. 19 (1966), 275–284.CrossRefGoogle Scholar
8.Krempa, J., Radicals of semigroup rings, Fundamenta Math. 85 (1974), 57–71.CrossRefGoogle Scholar
9.Pearson, K. R., Stephenson, W., and Watters, J. F., Skew polynomials and Jacobson rings, Proc. London Math. Soc. 42 (1981), 559–576.CrossRefGoogle Scholar
10.Puczylowski, E. R., Some questions concerning radicals of associative rings, Theory of Radicals, Coll. Math. Soc. J. Bolyai, 61, (Hungary 1991), 209–227.Google Scholar
11.Rowen, L. H., Polynomial identities in ring theory (Academic Press, 1980).Google Scholar