Commutativity of Periodic Rings with Some Identities in the Center

Abstract

Let R be an periodic ring. In this paper, we prove that an $(n+1)n$-tortion free periodic ring satisfying the properties $(ab)^{n}-ba \in Z(R) $, $(ab)^{n+1}- ba\in Z (R)$, $a^{n}(ab)-(ba)a^{n}\in Z(R)$ for all $a,b \in R$ is commutative, then R is commutative.