Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T10:29:07.270Z Has data issue: false hasContentIssue false
  • English
  • Français

Fragmented rings

Published online by Cambridge University Press:  14 November 2011

Paul Dubreil
Paul Dubreil
Affiliation:
Université de Paris VI
Get access Rights & Permissions [Opens in a new window]

Synopsis

The publication of the results in this paper has been delayed, for non-mathematical reasons. However, the author has given lectures on these results in Smolenice [2], Gainesville, Tulane. Riverside (1971), Milan (1972) and in Paris. The main consideration in this paper is the notion of a fragmented ring and its multiplicative semigroup. A fragmented ring is a ring with an identity having a finite set of idempotents, these commuting and therefore being central. In a subsequent paper we shall consider associated ideas in a purely semigroup-theoretic context and, in so doing, point out some differences between general semigroups and those semigroups that are the multiplicative semigroups of rings.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 78 , Issue 3-4 , 1978 , pp. 273 - 283
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Clifford, A. H. and Preston, G. B.. The algebraic theory of semigroups. Amer. Math. Soc. Surveys 7 (1961).Google Scholar
2 Dubreil, P.. Etude du demi-groupe multiplicatif de certains anneaux. Colloque de Smolenice (1968).Google Scholar
3 Foster, A. L.. The idempotent elements of a commutative ring form a Boolean algebra; ring-duality and transformation theory. Duke Math. J. 12 (1945), 143–152.CrossRefGoogle Scholar
4 Jacobson, N.. Structure of Rings. Colloquium Publ. 37 (Providence R.I.: Amer. Math. Soc, 1956).CrossRefGoogle Scholar
5 Parizek, B. and Schwartz, S.. On the multiplicative semigroup of residue classes (mod m). Mat. Casopis Sloven. Akad. Vied. 8 (1958), 136–150.Google Scholar
6 Posner, E. C.. Left valuation rings and simple radical rings. Trans. Amer. Math. Soc. 107 (1963), 458–465.CrossRefGoogle Scholar
7 Shin, G.. Prime ideals and sheaf representation of a pseudo-symmetric ring. Trans. Amer. Math. Soc. 184 (1973), 43–60.CrossRefGoogle Scholar
8 Waerden, B. L. van der. Algebra, 2 (Berlin: Springer, 1966).Google Scholar