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Keywords:
factorsubring; s-unital ring; commutativity; commutator; associative ring
factorsubring; s-unital ring; commutativity; commutator; associative ring
Summary:
Let $m \geq 0, ~r \geq 0, ~s \geq 0, ~q \geq 0$ be fixed integers. Suppose that $R$ is an associative ring with unity $1$ in which for each $x,y \in R$
there exist polynomials $f(X) \in X^{2} \mbox{$Z \hspace{-2.2mm} Z$}[X], ~g(X), ~h(X) \in X \mbox{$Z \hspace{-2.2mm} Z$}[X]$ such that $\{ 1-g (yx^{m}) \} [x, ~x^{r}y ~-~ x^{s}f (y x^{m}) x^{q}] \{ 1-h(yx^{m}) \} ~=~ 0$. Then $R$ is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of $x$ and $y$. Finally, commutativity of one sided s-unital ring is also obtained when $R$ satisfies some related ring properties.
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