On commutativity of associative rings with constraints involving a subset. (English) Zbl 0683.16025

Let R be a ring, N its set of nilpotent elements, and A a non-empty subset of R. Define R to have property (II-A) if for each non-central \(x\in R\), there exists a polynomial p(t)\(\in {\mathbb{Z}}[t]\) such that \(x- x^ 2p(x)\in A\); and let (*) be the property that for each \(x,y\in R\), there exist integers \(m=m(x,y)>1\) and \(n=n(x,y)\geq 1\) for which \([x,xy- y^ mx^ n]=0\). Theorem 1 asserts that R is commutative iff R has property (*) and also satisfies (II-A) for some commutative \(A\subseteq N\). Theorem 2 establishes that each of the following is equivalent to commutativity in left s-unital rings: (i) R has property (*) and there exists \(A\subseteq N\) for which R has property (II-A); (ii) for some fixed integer \(n\geq 1\), R has the property that for each \(y\in R\), there exists \(m=m(y)>1\) such that \([x,xy-y^ mx^ n]=[x,xy^ m-y^{m^ 2}x^ n]=0\) for all \(x\in R\). The authors give examples which preclude certain extensions of these results; and they conjecture that property (*) implies commutativity in rings with 1.
Reviewer: H.E.Bell


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