## On identities which are equivalent with commutativity.(English)Zbl 0538.16028

Let $$f(x,y)=\sum^{d}_{r=1}\sum^{r}_{i=0}f_{ri}(x,y)$$, where $$f_{ri}$$ denotes the sum of all terms of f with degree i in x and r-i in y, be a polynomial in two (noncommutative) indeterminates with integer coefficients. Find necessary and sufficient conditions on f such that a ring is commutative iff it satisfies the identity $$f=0$$. The author nearly completely solves the above question.
As is easily seen, we must have $$s_{ri}=0$$ for all r and i, where $$s_{ri}$$ denotes the sum of the coefficients of $$f_{ri}$$. Thus $$f(x,y)=m[x,y]+\sum^{d}_{r=3}\sum^{r-1}_{i=1}f_{ri}(x,y)$$ and $$m=\pm 1$$. A further condition assumed on f is that $$f_{r1}=0$$ for all r. The main result of the paper states that the above conditions on f are sufficient for equivalence with commutativity. It is also shown by way of an example that the first two conditions alone will not suffice.
Reviewer: Ş.Buzeţeanu

### MSC:

 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16Rxx Rings with polynomial identity 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)

### Keywords:

polynomial identity; commutativity