Trace-like functions on rings with no nilpotent elements

Let R be a ring with no nilpotent elements, with extended center C, and let E be the set of idempotents in C. Our first main result is that for any finite group G acting as automorphisms of R, there exist a finite set L C E and an /?c-bimodule homomorphism t: R -> (RL)G such that t(R) is an essential ideal of (RE)G. This theorem is applied to show the following: if R is a Noetherian, affine PI-algebra (with no nilpotent elements) over the commutative Noetherian ring A, and G is a finite group of A-automorphisms of R such that RG is Noetherian, then RG is affine over A.