Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-08T23:49:13.752Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

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 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