Let R be an associative ring satisfying the identity \(x^ ny^ n=y^ nx^ n\) and for any \(x\in R\), \(x\in Rx\cap xR\). The authors prove that R is commutative if one of the following conditions holds in R: i) \(n[x,y]=0\) implies \([x,y]=0\) and for any x,y\(\in R\) there exists a positive integer \(m=m(x,y)\) such that \((m,n)=1\), \([x,[x,(xy)^ m]]=0\); ii) R satisfies the identity \([x,[x,(xy)^ m]]=0\) and \(mn[x,y]=0\) implies \([x,y]=0\).