Strongly regular rings and rational identities of division rings (Q1092145)

A strongly regular ring may be defined as a ring R in which each element x has a `weak inverse' \(x_ 1\), satisfying \(x^ 2x_ 1=x\) and \(x^ 2_ 1x=x_ 1\). These equations determine \(x_ 1\) uniquely, so that \(x\mapsto x_ 1\) may be regarded as a unary operation on R and we can speak of the variety of strongly regular rings, and of strongly regular polynomials. The author points out that every strongly regular identity gives rise to a rational identity but not conversely, and he proves that with each rational polynomial p a strongly regular polynomial \(p^*\) can be associated such that p and \(p^*\) vanish identically on the same skew fields. Further he proves the following conditions equivalent, for any skew fields D, E. (a) D belongs to the variety of strongly regular rings generated by E, (b) D is isomorphic to a subring of an ultrapower of E, (c) any finite set of polynomials in \(Z<x_ 1,x_ 2,...>\) having a zero in D also has a zero in E. Moreover, when this holds, he shows that the PI-degree of D divides that of E.