Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T14:18:40.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Radical Rings

Published online by Cambridge University Press:  20 November 2018

W. Edwin Clark
W. Edwin Clark*
Affiliation:
University of Florida, Gainesville, Florida
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.

Let R be a ring. We denote by o the so-called circle composition on R, denned by a o b = a + b — ab for a, b ∊ R. It is well known that this composition is associative and that R is a radical ring in the sense of Jacobson (see 6) if and only if the semigroup (R, o) is a group. We shall say that R is a generalized radical ring if (R, o) is a union of groups. Such rings might equally appropriately be called generalized strongly regular rings, since every strongly regular ring satisfies this property (see Theorem A below). This definition was in fact partially motivated by the observation of Jiang Luh (7) that a ring is strongly regular if and only if its multiplicative semigroup is a union of groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 88 - 94
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

This research was supported by the National Science Foundation, Grant 216K07.

References

1. Albert, A. A., Structure of algebras, Amer. Math. Soc. Coll. Publ., 24 (1939).Google Scholar
2. Clark, W. E. and Lewin, J., On minimal ideals in the circle composition semigroup of a ring to appear in Publ. Math. Debrecen).Google Scholar
3. Clifford, A. H., Semigroups containing minimal ideals, Amer. J. Math., 70 (1948), 521–526.Google Scholar
4. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. I, Math Surveys, No. 7 (Amer. Math. Soc, 1961).Google Scholar
5. Drazin, M. P., Rings with central idempotent or nilpotent elements, Proc. Edinburgh Math. Soc, 9 (1958), 157–165.Google Scholar
6. Jacobson, N., Structure of rings, Amer. Math. Soc. Coll. Publ., 37 (1956).Google Scholar
7. Luh, J., A note on strongly regular rings, Proc. Japan Acad., 40, No. 2 (1964), 74–75.Google Scholar