Classification of rings satisfying some constraints on subsets

summary:Let $R$ be an associative ring with identity $1$ and $J(R)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ is a fixed positive integer and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$ for all $x,y\in R\backslash J(R)$ and (ii) $[x,[x,y^m]]=0$, for all $x,y\in R\backslash J(R)$. This result is also valid if (ii) is replaced by (ii)’ $[(yx)^mx^m-x^m(xy)^m,x]=0$, for all $x,y\in R\backslash N(R)$. Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14])