On weakly periodic-like rings and commutativity theorems

Abstract

A ring $R$ is called periodic if, for every $x$ in $R$, there exist distinct positive integers $m$ and $n$ such that $x^m=x^n$. An element $x$ of $R$ is called potent if $x^k=x$ for some integer $k>1$. A ring $R$ is called weakly periodic if every $x$ in $R$ can be written in the form $x=a+b$ for some nilpotent element $a$ and some potent element $b$ in $R$. A ring $R$ is called weakly periodic-like if every element $x$ in $R$ which is not in the center $C$ of $R$ can be written in the form $x=a+b$, with $a$ nilpotent and $b$ potent. Some structure and commutativity theorems are established for weakly periodic-like rings $R$ satisfying certain torsion-freeness hypotheses along with conditions involving some elements being central.