Generalized rational identities of power series rings (Q1086660)

Let A be a fixed algebra over a field F and let \(X=\{X_ 1,...,X_ m\}\) be noncommuting variables. Denote by R(X,A) the algebra of all formal rational expressions formed from \(X\cup A\). The ring P is an A-ring if A is a subring of P and the centre of P contains that of A. One says that \(f(X_ 1,...,X_ m)\in R(X,A)\) is a generalized rational identity (GRI) for P if \(f(p_ 1,...,p_ m)=0\) for all \((p_ 1,...p_ m)\in dom(f)\), the subset of \(P^ m\) where f is defined; f is non-degenerate if dom(f) is nonempty. Bergman proved that any two infinite dimensional division algebras with infinite centres have the same GRIs. The purpose of the paper under review is to generalize this result and to establish the following: Let \(P_ 1\) and \(P_ 2\) be non-GRI prime A-rings. Then any non-degenerate GRI of the power series ring \(P_ 1[[ x]]\) is a GRI for \(P_ 2[[ x]]\). An example is given, such that the sets of GRIs of \(P_ 1[[ x]]\) and \(P_ 2[[ x]]\) are different.