On rings with powers commuting on subsets (Q803250)

The authors call a ring R with 1 a (q,t,S)-ring if S is a nonempty subset of R and for every subset \(F\subseteq R\setminus S\) with \(| F| \leq q\), there exists a set Q of t pairwise relatively prime positive integers such that \([x^ i,y^ i]=0\) for all x,y\(\in F\), \(i\in Q\). It is shown that R is necessarily commutative if it satisfies one of the following conditions: (i) R is a (3,2,C)-ring, where C is a commutative ideal; (ii) R is a 2-torsion-free periodic (3,2,K)-ring, where K is a proper right ideal of R; (iii) R is a 2-torsion-free (4,4,K)-ring, where K is an additive subgroup of R and \(1\not\in K\); (iv) R is a (3,2,J(R))- ring, where J(R) is the Jacobson radical of R.