On rings with redundancy in multiplication
In this paper, the commutativity problem of rings is studied from the standpoint of a criterion of redundancy the introduction of which was stimulated by the work of \textit{G. A. Freiman} [Aequationes Math. 22, 140-152 (1981; Zbl 0489.20020)]. Let \(S\) be an arbitrary finite subset of the ring \(R\), \(S^ n=\{x_ 1\cdots x_ n|\) \(x_ i\in S\}\), \(n>1\), \(C(R)\) the commutator ideal of \(R\). The set \(S\) is called \(n\)-redundant if \(| S^ n| <| S|^ n\), and \(R\) is \((m,n)\)-redundant if every subset \(S\subset R\) with \(| S| =m\) is \(n\)-redundant. The main theorem asserts that any \((2,2)\)-redundant ring with a unit is commutative. The authors also show that \(C(R)\) is a nil-ideal if \(R\) is any \((m,n)\)-redundant algebra over an infinite field or an arbitrary \((2,3)\)- or \((3,2)\)-redundant ring. Proceeding from different assumptions the last result for rings was obtained by the authors and \textit{T. Jenkyns} and \textit{C. Laywine} [Result. Math. 13, 12-22 (1988; Zbl 0646.16026)].
scientific article; zbMATH DE number 4087870