On the mapping xy→(xy)n in an associative ring

We consider the following condition (*) on an associative ring R:(*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R2, and f(xy)=(xy)n(x,y) for some positive integer n(x,y)>1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for non-Artinian rings. The results generalize commutativity theorems found elsewhere. The case n(x,y)=2 is examined in detail

© Hindawi Publishing Corp. ON THE MAPPING xy → (xy) n IN AN ASSOCIATIVE RING

We consider the following condition (*) on an associative ring R: (*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R 2,and f(xy) = (xy) n(x,y) for some positive integer n(x,y)> 1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for non-Artinian rings. The results generalize commutativity theorems found elsewhere. The case n(x,y) = 2 is examined in detail. 2000 Mathematics Subject Classification: 16D70, 16P20. Let R be an associative ring, not necessarily with unity, and let R + denote the additive group of R. In[3], it was shown that R is commutative if it satisfies the following condition. (I) For each x and y in R, there exists n = n(x,y)> 1 such that (xy) n = xy. We generalize this result by considering the condition below. (II) There exists a function f from R into R such that f is a group homomorphism of R +, f is injective on R 2,andf(xy) = (xy) n(x,y) for some positive integer n = n(x,y)> 1 depending on x and y