Commutativity of rings through a Streb's result

Abstract

In this paper we investigate commutativity of rings with unity satisfying any one of the properties:

$$\left\{ {{\text{1 - }}g\left( {yx^m } \right)} \right\}\left[ {yx^m - x^r f\left( {yx^m } \right)x^s ,x} \right]\left\{ {1 - h\left( {yx^m } \right)} \right\} = 0$$

$$\left\{ {{\text{1 - }}g\left( {yx^m } \right)} \right\}\left[ {x^m y - x^r f\left( {yx^m } \right)x^s ,x} \right]\left\{ {1 - h\left( {yx^m } \right)} \right\} = 0$$

$$y^t \left[ {x,y^n } \right] = g\left( x \right)\left[ {f\left( x \right),y} \right]h\left( x \right){\text{ and }}\left[ {x,y^n } \right]y^t = g\left( x \right)\left[ {f\left( x \right),y} \right]h\left( x \right)$$

for some f(X) in $X^2 \mathbb{Z}\left[ X \right]$ and g(X), h(X) in $\mathbb{Z}\left[ X \right]$ where m ≥ 0, r ≥ 0, s ≥ 0, n > 0, t > 0 are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements x and y for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize a number of commutativity theorems established recently.