## 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

### MSC:

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

### Keywords:

commutativity; commutators