Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T09:48:41.987Z Has data issue: false hasContentIssue false
  • English
  • Français

The Commutativity of a Special Class of Rings

Published online by Cambridge University Press:  20 November 2018

Wallace S. Martindale III
Wallace S. Martindale III*
Affiliation:
University of Pennsylvania Yale University
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.

A well-known theorem of Jacobson (1) states that if every element x of a ring R satisfies xn(x) = x where n(x) > 1 is an integer, then R is commutative. A series of generalizations of this theorem have been proved by Herstein (2; 3; 4; 5; 6), his last result in this direction (6) being that a ring R is commutative provided every commutator u of R satisfies un(u) = u. We now define a γ-ring to be a ring R in which un(u) — u is central for every commutator u of R (where n(u) > 1 is an integer). In the present paper we verify the following conjecture of Herstein: every commutator of a γ-ring is central.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 263 - 268
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Jacobson, N., Structure theory for algebraic algebras of bounded degree, Ann. Math., 46 (1945), 695–707.Google Scholar
2. Herstein, I.N., A generalization of a theorem of Jacobson, Amer. J. Math., 73 (1951), 756–762.Google Scholar
3. Herstein, I.N., A generalization of a theorem of Jacobson II, Abstract, Bull. Amer. Math. Soc, 58 (1952), 383.Google Scholar
4. Herstein, I.N., A generalization of a theorem of Jacobson III, Amer. J. Math., 75 (1953), 105–111.Google Scholar
5. Herstein, I.N., The structure of a certain class of rings, Amer. J. Math., 75 (1953), 864–871.Google Scholar
6. Herstein, I.N., A condition for the commutativity of rings, Can. J. Math., 9 (1957), 583–586.Google Scholar
7. Krasner, M., The non-existence of certain extensions, Amer. J. Math., 75 (1953), 112–116.Google Scholar
8. Schilling, O.F.G., The theory of valuations, Math. Surveys IV (New York, 1950).Google Scholar
9. Weyl, H., Algebraic theory of numbers (Princeton, 1940).Google Scholar