SOME COMMUTATIVITY THEOREMS FOR ASSOCIATIVE RINGS WITH CONSTRAINTS INVOLVING A NIL SUDSET

We first prove that a ring $R$ with unity 1 is corrunutalive if and only if for each $x$ in $R$ either $x$ is central or there exists a polynomial $f(t) \in Z[t]$ such that $x- x^2f(x) \in A$, where $A$ is a nil subset of $R$ (not necessarily a subring of $R$) and $R$ stisfies any one of the conditions $[x, x^my- x^py^nx^q] =0$ and $[x,yx^m-x^Py^nx^q]=0$ for all $x,y$ in $R$, where $m\ge 0$, $n >1$, $p \ge 0$, $q \ge 0$ are integers depending on pair of elements $x$, $y$. Further the same result has been extended for one sided $s$-unital rings. Finally a related result for a nil commutative subset $A$ is also obtained.