Functional identities in upper triangular matrix rings. (Q905746)

Let \(R\) be a subring of an associative ring \(Q\); one requires that \(R\) and \(Q\) share the same unit element. Denote \(\overline x_m=(x_1,\ldots,x_m)\in R^m\) and let \(\overline x_m^i\) be the ``vector'' \(\overline x_m\) without its \(i\)-th coordinate, analogously \(\overline x_m^{ij}\) stands for the same vector picking out its coordinates \(i\) and \(j\). Denote by \(T_n(R)\) the ring of upper triangular \(n\times n\) matrices over \(R\). The main contribution of the paper under review is the following theorem. Let \(R\) be a \(d\)-free subset of \(Q\), then \(T_n(R)\) is a \(d\)-free subset of \(T_n(Q)\) for each \(n\). Although the statement seems quite natural and transparent its proof is rather sophisticated and lengthy.