 By Nilolaus Vonessen

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Here the second equality holds because RG is a finite CG-module. And the last GKdimension is by assumption independent of the choice of P G Min (R). It follows that there are no strict inclusion relations among the primes in (MinC) n CG. Since the latter set clearly contains MinC G , it follows that (MinC) n CG = MmCG. 9 applies to the action of G on C. 14 does not hold if R is not a finite module over its center. We will need here a technical result which we will prove only in §8. 15 EXAMPLE. Let G be a connected linearly reductive group.

ACTIONS ON PI-ALGEBRAS 43 Now let Q G Min (C) be a minimal prime of C. Choose a minimal prime P of R such that P D C = Q. Then QC\CG = Pn CG, and GK(C G /(Q n CG)) = GK(CG/{P = n CG)) GK(RG/(PnRG)). Here the second equality holds because RG is a finite CG-module. And the last GKdimension is by assumption independent of the choice of P G Min (R). It follows that there are no strict inclusion relations among the primes in (MinC) n CG. Since the latter set clearly contains MinC G , it follows that (MinC) n CG = MmCG.

45 NlKOLAUS VONESSEN 46 If B is affine, it has a finite number of minimal prime ideals, and its prime radical is nilpotent (concerning the latter, see [Braun 84]). In order to prove the lemma, we may clearly factor out by a nilpotent ideal of B. Hence it suffices to prove the lemma under the assumption that B is semiprime and that B has only a finite number of minimal prime ideals. PROOF. Denote by Z the center of P, and by K the total ring of fractions of Z. 7]. Denote by KA the subalgebra of KB generated by K and A.

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