Chacron’s condition and commutativity theorems. (English) Zbl 0705.16023

Let R denote a ring. Chacron’s condition is condition (C): for every x,y\(\in R\), there exist \(f(X),g(X)\in X^ 2{\mathbb{Z}}[X]\) such that \([x- f(x),y-g(y)]=0\). Theorem 1 provides several characterizations of rings with condition (C) satisfying an identity of the form \([X^ n,Y^ n]=0\); Theorem 2 gives several extended-commutator conditions each of which in conjunction with (C) implies commutativity of R; the remaining theorems identify strong versions of (C) which imply commutativity. Extensive use is made of recent structural results for non-commutative rings, due to W. Streb [Math. J. Okayama Univ. 31, 135-140 (1989; Zbl 0702.16022)] - results which the present authors state and prove in the first section.
Reviewer: H.E.Bell


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)


Zbl 0702.16022