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

Kezlan, Thomas P.

Math. Jap. 29, 135-139 (1984).

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