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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
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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

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