COMMUTATIVITY OF RIGHT $S$-UNITAL RINGS UNDER SOME POLYNOMIAL CONSTRAINTS

In the present paper we discuss the commutativity of certain rings, namely rings with unity 1 and right s-unital rings under each of the following conditions: \[ (P1)[yx^m - x^nf (y), x] = 0, \quad (P1)^*[yx^m - f (y)x^n, x] = 0, \]where $m$, $n$ are fixed non-negative integers and $f(x)$ is a polynomial in $X^2\mathbb{Z}(X)$ varying with the pair of ring elements $x$, $y$. Further, the results have been extended to the case when $m$ and $n$ depend on the choice of $x$ and $y$ and the ring satisfies the Chacron's condition.