Hostname: page-component-6bf8c574d5-mggfc Total loading time: 0 Render date: 2025-02-23T02:57:56.986Z Has data issue: false hasContentIssue false
  • English
  • Français

On semigroups of endomorphisms of generalized Boolean rings

Published online by Cambridge University Press:  09 April 2009

Douglas B. Smith Jr  and
Jiang Luh
Douglas B. Smith Jr
Affiliation:
USAF Academy, Colorado 80840, U.S.A.
Jiang Luh
Affiliation:
North Carolina State University at Raleigh 27607, U.S.A.
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.

Magill in [4] first proved that two Boolean rings are isomorphic if and only if their respective endomorphism semigroups are isomorphic. His proof, however, relied on topological techniques. More recently Maxson has published a proof of the above using purely algebraic techniques [5]. In this paper, structure theorems are given which allow us to extend the above result to the pk-rings of Foster [1]. As a special case, the result is shown to apply also to p-rings. An example is given to show that a further extension to J-rings is impossible.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 4 , December 1974 , pp. 411 - 418
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Foster, A. L., ‘pk-rings and ring-logics’, Ann. Scu. Norm. Pisa 5 (1951), 279–300.Google Scholar
[2]Foster, A. L., ‘Generalized ‘Boolean’ theory of universal algebras, PartI' Math. Z., 58 (1953) 306–336.Google Scholar
[3]Jacobson, N., Structure of Rings. (Amer. Math. Soc. Colloquim Vol. 36, revised edition (1964)).Google Scholar
[4]Magill, K. D. Jr, ‘The Semigroup of endomorphisms of a Boolean ring’, J. Austral. Math. Soc. 11 (1970), 411–416.CrossRefGoogle Scholar
[5]Maxson, C. J., ‘On semigroups of Boolean ring endomorphisms’, Semigroup Forum 4 (1972), 78–82.CrossRefGoogle Scholar
[6]McCoy, N. H., ‘Subrings of Direct Sums’, Amer. J. Math. 60 (1938), 374–382.CrossRefGoogle Scholar
[7]McCoy, N. H., Rings and Ideals. (Carus Math. Monographs Vol. 8 (1948)).CrossRefGoogle Scholar