P.I. G-rings and the contractability of primes

Let R=F{x ₁, ..., x_k} be a prime affine p.i. ring and S a multiplicative closed set in the center of R, Z(R). The structure of G-rings of the form R _s is completely determined. In particular it is proved that Z(R _s)-the normalization of Z(R _s) -is a prüfer ring, 1≦k.d(R _s)≦p.i.d(R _s) and the inequalities can be strict. We also obtain a related result concerning the contractability of q, a prime ideal of Z(R) from R. More precisely, let Q be a prime ideal of R maximal to satisfy QΥ{hooked}Z(R)=q. Then k.d Z(R)/q=k.d R/Q, h(q)=h(Q) and h(q)+k.d Z(R)/q=k.d z(R). The last condition is a necessary but not sufficient condition for contractability of q from R.