TL;DRAbstract
Let p, q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s-unital ring satisfying $$[f(x^p y^q )\; - x^r y,x]\; = \;0\;\;([f(x^p y^q )\; - yx^r ,x] = 0$$ , respectively) where $$f(\lambda ) \in \lambda ^2 \mathbb{Z}\;[\lambda ]$$ , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.
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Let p, q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s-unital ring satisfying $$[f(x^p y^q )\; - x^r y,x]\; = \;0\;\;([f(x^p y^q )\; - yx^r ,x] = 0$$ , respectively) where $$f(\lambda ) \in \lambda ^2 \mathbb{Z}\;[\lambda ]$$ , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.
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