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On the Commutativity of a Ring with Identity
Published online by Cambridge University Press: 20 November 2018
Abstract
Let R be a ring with identity. R satisfies one of the following properties for all x, y ∈ R:
(I) xynxmy = xm+1yn+1 and mnm! n! x≠0 except x = 0;
(II) xynxm = xm + 1yn + 1 and mm! n! x≠0 except x = 0;
(III) xmyn = ynxm and m! n! x≠0 except x = 0;
(IV) (xpyQ)n = xpnyqn for n = k, k + 1 and N(p, q, k) x≠0 except x = 0, where N(p, q, k) is a definite positive integer. Then R is commutative.
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- Copyright © Canadian Mathematical Society 1984
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