A commutativity theorem for rings. II. (English) Zbl 0575.16017

[Part I, cf. Math. J. Okayama Univ. 26, 109-111 (1984; Zbl 0568.16017).]
The author generalizes a theorem for commutativity of a ring given by T. P. Kezlan [Math. Jap. 29, 135-139 (1984; Zbl 0538.16028)]. This generalization is the following theorem: ”Let \(n>0\), r,s, and t be nonnegative integers and let \(f(X,Y)=\sum^{r}_{i=1}\sum^{s}_{j=2}f_{ij}(X,Y)\) be a polynomial in two noncommuting indeterminates X, Y with integer coefficients such that each \(f_{ij}\) is a homogeneous polynomial with degree i in X and degree j in Y and the sum of the coefficients of \(f_{ij}\) equals zero. Suppose a left s-unital ring R satisfies the polynomial identity \(X^ t[X^ n,Y]-f(X,Y)=0\). If either \(n=1\) or \(r=1\) and R has the property Q(n), then R is commutative.”
Reviewer: M.Ştefănescu


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16Rxx Rings with polynomial identity