Some notes on \(CN\) rings. (Q745943)

Let \(R\) be a ring with 1, and denote by \(Z(R)\) and \(N(R)\) respectively the center of \(R\) and the set of nilpotent elements of \(R\). Call \(R\) a \(CN\)-ring if \(N(R)\subseteq Z(R)\). The initial theorems give equivalent conditions for \(R\) to be a \(CN\)-ring, among them that there exists a positive integer \(n\) such that for each \(x\in N(R)\) and \(y\in R\), \(((1+x)y)^{n+k}=(1+x)^{n+k}y^{n+k}\) for \(k=0,1,2\). Most of the remaining theorems deal with commutativity of rings satisfying conditions of the form \[ (xy)^{n+k}=x^{n+k}y^{n+k}\text{ for fixed }n,\text{ and }k= 0,1,2,\tag{*} \] and certain choices of \(x\) and \(y\). These results are motivated by a theorem of \textit{S. Ligh} and \textit{A. Richoux} [Bull. Aust. Math. Soc. 16, 75-77 (1977; Zbl 0335.16034)] which asserts that \(R\) must be commutative if there exists \(n\) such that (*) holds for all \(x,y\in R\).