Weakly periodic and subweakly periodic rings. (Q1811972)

Let \(R\) be a ring, \(N\) the set of nilpotent elements, \(J\) the Jacobson radical, and \(P= \{x\in R\mid\) there exists an integer \(n= n(x)> 1\) such that \(x^n= x\}\). The ring \(R\) is called periodic if for each \(x\in R\), there exist distinct positive integers \(m\), \(n\) for which \(x^n= x^m\); and \(R\) is called weakly periodic if \(R= P+ N\). It is known that every periodic ring is weakly periodic; whether the converse holds is an open question. The authors generalize further, defining \(R\) to be subweakly periodic if \(R\setminus J\subseteq P+ N\); and they prove commutativity theorems for subweakly periodic rings. Their principal result establishes commutativity of subweakly periodic rings satisfying the following two conditions: (i) \(J\) is commutative; (ii) for each \(x, y\in R\), there exist words \(w(x, y)\) and \(w'(x,y)\) in \(x\) and \(y\), for which \(w(x, y)[xy, yx]= 0= [xy, yx]w'(x, y)\).