Rings graded by polycyclic-by-finite groups

We use the duality between group gradings and group actions to study polycyclic-by-finite group-graded rings. We show that, for such rings, graded Noetherian implies Noetherian and relate the graded Krull dimension to the Krull dimension. In addition we find a bound on the length of chains of prime ideals not containing homogeneous elements when the grading group is nilpotent-by-finite. These results have suitable corollaries for strongly groupgraded rings. Our work extends several results on skew group rings, crossed products and group-graded rings. Introduction. An associative ring with identity is said to be graded by the group G if R = E R(x) xEG is a direct sum of additive subgroups R(x) with R(x)R(y) C R(xy). It follows that 1R E R(1). R is said to be strongly G-graded if R(x)R(y) = R(xy) for all x, y E G. The group G is said to be polycyclic-by-finite if G has a subnormal series 1 = Go a G, a * * a Gn = G, where Gi/Gi-, is either infinite cyclic or finite. If G has a normal series of this type, then G is said to be strongly polycyclic-by-finite. The number of infinite cyclic factors which occur in this series is called the Hirsch number of G and is denoted by h(G). Since any two series have isomorphic refinements, h(G) is a nonzero integer invariant of G. In their paper [6] M. Cohen and S. Montgomery proved a duality theorem for finite group gradings and group actions on rings. Their methods provided a way to translate results on finite crossed products to more general group-graded rings. Their construction was developed concretely by the second author in [11], where further applications were given. In addition the approach there works for infinite groups. Here we use this method to study Krull dimension and chains of prime ideals in rings graded by polycyclic-by-finite groups. The construction of [11] for infinite groups is sketched in ?1. We comment that the duality theorem of Cohen and Montgomery has been extended by R. Blattner and S. Montgomery [4] to handle certain infinite groups as a corollary to a more general theorem on Hopf algebra actions. Other studies of the duality have been made by M. van den Bergh [2] and J. Osterburg [10]. We now describe the main results of this paper. Let R be graded by a polycyclicby-finite group G. In ?2 we show that a graded R-module M, which is graded Noetherian, is in fact Noetherian. Also in this situation, we show that the Krull Received by the editors October 7, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A03, 16A55, 16A33; Secondary 16A24. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page