Commutativity of rings with some commutator constraints. (English) Zbl 0697.16031

Bell, H. E.; Quadri, M. A.; Ashraf, M.

Rad. Mat. 5, No. 2, 223-230 (1989).

In the present paper the authors continue their investigations on the commutativity of rings satisfying some conditions on the commutators. Let R be an associative ring with 1, Z the ring of integers and x and y noncommuting variables. The commutativity of R is established in the following cases.
1. R satisfies the identity \([xy-p(yx),x]=0\) where \(p(t)\in t^ 2Z[t].\)
2. For each \(x,y\in R\) there exists p(t)\(\in tZ[t]\) such that \([xy,x](p(xy)-1)=0.\)
3. For each \(x,y\in R\) there exist \(p(t),q(t)\in t^ 2Z[t]\) such that \([xy-p(yx),x]=[xy-q(yx),y]=0.\)
4. There exists a fixed integer \(n\geq 1\) and for each \(y\in R\) there exists an integer \(m=m(y)>1\) for which \([x,xy-x^ ny^ m]=0\) for every \(x\in R.\)
5. For every \(x,y\in R\) there exist integers \(n=n(x,y)>1\) and \(m=m(x,y)>1\) such that \([xy-y^ nx,x]=[xy-yx^ m,y]=0\).

Reviewer: P.Koshlukov

MSC:

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

Keywords:

commutativity; commutators