ABSTRACT Let k be an algebraically closed field, and let R be an affine (i.e., finitely gener- ated) A -algebra satisfying a polynomial identity. Let G be a linearly reductive group acting rationally on R. In this paper, the relationship between R and the fixed ring RG is studied. Best results have been obtained if R is left Noetherian, or even an Azumaya algebra, or if G acts by inner automorphisms. Among the results for left Noetherian algebras are the following, (a) The fixed ring RG is affine — this is an extension of Hilbert's famous theorem for commutative algebras, (b) "Lying over" holds. That is, given a prime ideal p of RG, there is a prime ideal P of R such that p is a minimal prime over P D RG. (c) Further results concern localization. E.g., if R is prime, then RG has a total ring of fractions which is Artinian and which is contained in the total ring of fractions of R. This means in particular that the regular elements of RG are also regular in R. These and other results actually characterize linearly reductive groups: If G is a linear algebraic group which is not linearly reductive, then a rational action of G on an affine prime Noetherian Pi-algebra R is constructed such that RG is neither affine nor Noetherian, and lying over does not hold. This is an important difference to commutative invariant theory where in prime characteristic most results can be proven for reductive groups. If one, however, assumes that R is a finite module over its center, then the above properties hold in prime characteristic also for actions of reductive groups. Finally, the question is studied whether and when one can define a "map" from the prime spectrum of R to the spectrum of i£G, and what the obstacles are. Key words and phrases. Affine Pi-algebras, linearly reductive groups, group actions, trace ring, restricted extensions of Noetherian rings, intermediate centralizing extensions, Azumaya algebras, lying over, inner automorphisms. Received by the editors June 22,1988 IV

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