Some commutativity theorems for right \(s\)-unital rings. (English) Zbl 0767.16010

Let \(n\), \(m\), \(r\), \(s\) and \(t\) be fixed non-negative integers. For a ring \(R\) we consider the following polynomial identity: (1) \([x^ n,y]x^ t=y^ r[x,y^ m]y^ s\) for all \(x,y\in R\). In this paper it is proved that each of the following conditions implies the commutativity of the rings \(R\) which satisfy the polynomial identity (1): i) \(n>1\) and \(R\) is a right \(s\)-unital ring which possesses the property \(Q(n)\) (for all \(x,y\in R\), \(n[x,y]=0\) implies \([x,y]=0\)); ii) \(n > 1\) and \(R\) is a semiprime ring; iii) \(m > 1\), \(n > 1\), \(m\) and \(n\) relatively prime integers, and \(R\) is a right \(s\)-unital ring; iv) \(n=1\), \((t,m,r,s)\neq (0,1,0,0)\) and \(R\) is an \(s\)-unital ring. These results improve some theorems established by the first author [PU. M. A., Pure Appl. Math., Ser. A 1, No. 2, 97-108 (1990; Zbl 0731.16022)].
Reviewer: M.Guţan (Iaşi)


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


Zbl 0731.16022