Commutativity of associative rings through a Streb’s classification. (English) Zbl 0913.16017

Ashraf, Mohammad

Arch. Math., Brno 33, No. 4, 315-321 (1997).

Some sufficient conditions for an associative ring to be commutative are presented. The symbol \([x,y]\) denotes the commutator \(xy-yx\) and \(\mathbb{Z}\) stands for the ring of integers. If \(R\) is a ring with unity such that for each \(x,y\in R\) there exist polynomials \(f(x)\in x^2\mathbb{Z}[x]\) and \(g(x),h(x)\in x\mathbb{Z}[x]\) such that \(\{1-g(yx^m)\}[x,x^r-x^sf(yx^m)x^q]\{1-h(yx^m)\}=0\) for some non-negative integers \(m,r,s,q\), then \(R\) is commutative (Theorem 1). If \(R\) is a left \(s\)-unital ring (\(x\in Rx\) for all \(x\in R\)) such that for each \(x,y\in R\) there exists a polynomial \(f(x)\in x^2\mathbb{Z}[x]\) such that \([x^ry-x^sf(yx^m)x^q,x]=0\) for some non-negative integers \(m,r,s,q\), then \(R\) is commutative (Theorem 3).

Reviewer: Ladislav Bican (Praha)

MSC:

16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)

Keywords:

\(s\)-unital rings; commutativity theorems; commutators