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A commutativity theorem for power-associative 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 a power-associative ring with identity and let I be an ideal of R such that R/I is a finite field and x ≡ y (mod I) implies x2 = y2 or both x and y commute with all elements of I. It is proven that R must then be commutative. Examples are given to show that R need not be commutative if various parts of the hypothesis are dropped or if “x2 = y2” is replaced by “xk = yk” for any integer k > 2.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 1 , August 1970 , pp. 75 - 79
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
[2] Outcalt, D.L. and Yaqub, Adil, “A commutativity theorem for rings”, Bull. Austral. Math. Soc. 2 (1970), 95–99.CrossRefGoogle Scholar