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A commutativity theorem for rings

Published online by Cambridge University Press:  17 April 2009

D.L. Outcalt  and
Adil Yaqub
D.L. Outcalt
Affiliation:
University of California, Santa Barbara, California.
Adil Yaqub
Affiliation:
University of California, Santa Barbara, California.
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Abstract

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Let R be an associative ring with identity in which every element is either nilpotent or a unit. The following results are established. The set N of nilpotent elements in R is an ideal. If R/N is finite and if x ≡ y (mod N) implies x2 = y2 or both x and y commute with all elements of N, then R is commutative. Examples are given to show that R need not be commutative if “X2 = y2” is replaced by “xk = yK” for any integer k > 2. The case N = (0) yields Wedderburn's Theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 1 , February 1970 , pp. 95 - 99
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Outcalt, D.L. and Yaqub, Adil, “A generalization of Wedderburn's theorem”, Proc. Amer. Math. Soc. 18 (1967), 175–177.Google Scholar