International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 3, Pages 513-517
doi:10.1155/S0161171284000569
Commutativity theorems for rings and groups with constraints on commutators
Department of Mathematics, University of Thessaloniki, Thessaloniki, Greece
Received 5 April 1983; Revised 10 April 1984
Copyright © 1984 Evagelos Psomopoulos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let n>1, m, t, s be any positive integers, and let R be an associative ring with identity. Suppose xt[xn,y]=[x,ym]ys for all x, y in R. If, further, R is n-torsion free, then R is commutativite. If n-torsion freeness of R is replaced by “m, n are relatively prime,” then R is still commutative. Moreover, example is given to show that the group theoretic analogue of this theorem is not true in general. However, it is true when t=s=0 and m=n+1.