COMMUTATIVITY THEOREMS FOR RINGS WITH CONSTRAINTS ON COMMUTATORS

H. A. S. ABUJABAL, MOHD ASHRAF
1995 Tamkang Journal of Mathematics  
Let $R$ be a left (resp. right) $s$-unital ring and $m$ be a positive integer. Suppose that for each $y$ in $R$ there exist $J(t)$, $g(t)$, $h(t)$ in $Z[t]$ such that $x^m[x,y]= g(y)[x,y^2f(y)]h(y)$ (resp. $[x,y]x^m= g(y)[x,y^2f(y)]h(y))$ for all $x$ in $R$. Then $R$ is commutative (and conversely). Finally, the result is extended to the case when the exponent $m$ depends on the choice of $x$ and $y$.
doi:10.5556/j.tkjm.26.1995.4375 fatcat:55bbgm4me5dkdojidukaof5a6u