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

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


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