On commutativity of rings. (English) Zbl 0718.16031

Several theorems in this paper establish commutativity of rings satisfying constraints of the form \([x,x^ my-f(y)x^ n]=0\), sometimes in conjunction with other conditions. For example, it is proved that a ring R is commutative if for each x,y\(\in R\), there exist integers \(m>0\) and \(n\geq 0\) and \(f(t),g(t),h(t)\in t^ 2{\mathbb{Z}}[t]\) with \(f(1)=\pm 1\) such that \([x,x^ my-f(y)x^ n]=0=[x-g(x),y-h(y)].\)
There is also a theorem asserting commutativity of certain left s-unital rings satisfying polynomial identities of the form \(X^ m[f(X),Y]+v(X,Y)[X,g(Y)]w(X,Y)=0\), where f(X),g(X)\(\in X{\mathbb{Z}}[X]\), and v(X,Y) and w(X,Y) are monic monomials in noncommuting indeterminates X and Y; and there is a new proof of a recent result of the reviewer [Result. Math. 18, 197-201 (1990)].


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
16R40 Identities other than those of matrices over commutative rings