On weakly periodic-like rings and commutativity. (Q867488)

Let \(R\) be a ring; and let \(Z\), \(J\), \(E\) and \(N\) denote respectively the center, the Jacobson radical, the set of idempotents, and the set of nilpotent elements. Let \(p\) be a fixed prime. Define \(R\) to be an \(N_0\)-ring if for each \(x,y\in R\setminus(N\cup J\cup Z)\) there exists \(n=n(x,y)>1\) for which \(x^ny-xy^n\in N\); and if, in addition, \(pR=\{0\}\) and all \(n(x,y)\) can be taken as \(p\), call \(R\) a generalized \(p\)-ring. The author studies commutativity in \(N_0\)-rings and generalized \(p\)-rings with additional properties. A typical theorem states that a generalized \(p\)-ring with 1 must be commutative if \(J\) is commutative and \(E\subseteq Z\).