ON A THEOREM OF HERSTEIN DEDICATED TO PROFESSOR SHIH-TONG TU ON HIS 60TH BIRTHDAY

Abstract

Let $R$ be an associative ring with identity such that for some fixed integer $m >1$, $(x+y)^m =x^m+y^m$ for all $x,y$ in $R$. If $m =2$ (mod 4) ,or $p-1|m-1$ for each prime factor $p$ of $m$, then $R$ is commutative. The restriction on $m$ is essential. Moreover, in case of $m=2$ (mod 4) and $m >2$, then $R$ is isomorphic to a subdirect sum of subdirectly irreducible rings $R_i$ each of which, as homomorphic images of $R$, satisfies the same polynomial identity $(x +y)^m =x^m +y^m$; and for each $x$ in $R_i$;,either $x^2 =0$ or $x^{2q} =1$, where $(q,m)=1$.