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

Abujabal, Hamza A. S.; Obaid, Mustafa A.

Math. Jap. 37, No. 3, 591-600 (1992).

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)

MSC:

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

Keywords:

polynomial identity; commutativity; right \(s\)-unital ring; semiprime ring

Citations:

Zbl 0731.16022