## 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

### MSC:

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