On commutativity of rings with constraints on subsets. (English) Zbl 0808.16037

Abujabal, H. A. S.; Khan, M. A.; Khan, M. S.; Samman, M. S.

Czech. Math. J. 43, No. 3, 439-449 (1993).

The authors state thirteen ring properties of familiar form, most being constraints on commutators; and they prove that various combinations of these imply commutativity in appropriate classes of rings. The results are unsurprising extensions of work of several authors. A sample theorem is as follows: Let \(R\) be a ring and \(A\) a commutative set of nilpotent elements, and suppose that for each non-central \(x \in R\) there exists \(f(t) \in \mathbb{Z}[t]\) such that \(x - x^ 2 f(x) \in A\). If for each \(x, y \in R\) there exist integers \(k = k(x,y) \geq 1\), \(m = m(x,y) > 1\) and \(n = n(x,y) \geq 1\) such that \([x, x^ n y - y^ m x^ k] = 0\), then \(R\) is commutative.

Reviewer: H.E.Bell (St.Catherines)

MSC:

16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)
16N40	Nil and nilpotent radicals, sets, ideals, associative rings
16R40	Identities other than those of matrices over commutative rings

Keywords:

commutator constraints; commutativity; nilpotent elements

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