Commutativity theorems for rings and groups with constraints on commutators (Q2266053)

Let n and m be positive integers, let s and t be nonnegative integers, and let R be a ring with 1 which satisfies the identity \(x^ t[x^ n,y]=[x,y^ m]y^ s\). It is proved that R must be commutative if either of the following holds: (i) \(n>1\) and R is n-torsion-free: (ii) n and m are relatively prime. Condition (i) relates to earlier work of the author, \textit{H. Tominaga}, and \textit{A. Yaqub} [Math. J. Okayama Univ. 23, 37-39 (1981; Zbl 0474.16026)]; and the case \(s=t=0\) and \(m=n>1\) recovers a theorem of the reviewer [Can. Math. Bull. 21, 399-404 (1978; Zbl 0403.16024)]. The author also shows that a multiplicative group G must be abelian if it satisfies the identity \([x^ n,y]=[x,y^{n+1}]\) for some positive integer n.