Commutativity conditions on rings

Author:

Misso Paola

Abstract

We prove the following result: let R be an arbitrary ring with centre Z such that for every x, yR, there exists a positive integer n = n(x, y) ≥ 1 such that (xy)nynxnZ and (yx)nxnynZ; then, if R has no non-zero nil ideals, R is commutative. We also prove a result on commutativity of general rings: if R is r!-torsion free and for all x, yR, [xr, ys] = 0 for fixed integers rs ≥ 1, then R is commutative. As a corollary we obtain that if R is (n + 1)!-torsion free and there exists a fixed n ≥ 1 such that (xy)nynxn = (yx)nxnynZ for all x, yR, then R is commutative.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Reference3 articles.

1. Power maps in rings;Herstein;Michigan Math. J.,1961

2. Commutativity results for rings

3. Some commutativity results for rings;BeIl;Bull. Austral. Math. Soc.,1980

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