Some commutativity theorems through a Streb\'s classification

Abstract

In the present paper we investigate commutativity of rings with unity satisfying any one of the properties ${1-(x^my)g(x^my)}[x^my-x^rf(x^my)x^s,x]{1-(x^my)h(x^my)}=0$, ${1-(x^my)g(x^my)}[yx^m-x^rf(x^my)x^s,x]{1-(x^my)h(x^my)}=0$, $x^t[x^k,y]=g(y)[x,f(y)]h(y)$ and $[x^k,y]x^t=g(y)[x,f(y)]h(y)$ for some $f(X)$ in $X²Z[X],$ where $m≥ 0, r≥ 0, s≥ 0, k>0, t>0$ are non-negative integers. Finally, under different appopriate constraints on commutators, commutativity of R has been established.