Prime Ideals in Group Algebras Oe Polycyclic-by-finite Groups

Introduction Group algebras K[G] of poly cyclic-by-finite groups are easily-defined, interesting examples of right and left Noetherian rings. Since prime rings and prime ideals are the basic building blocks in the Goldie theory of Noetherian rings, the determination of the structure of the prime ideals of K[G] is certainly of importance. In a recent fundamental paper [8], Roseblade proved that G has a characteristic subgroup Go of finite index such that the prime ideals of K[G0] c a n De described in a particularly nice manner. Furthermore, he showed that this type of description does not, in general, apply to K[G]. In this paper, we offer a slightly different, somewhat more complicated formulation which does indeed describe the primes of K[G]. Observe that the group algebra K[G] can be written as K[G] = K[G0] * (G/Go), a crossed product of the finite group H = G/Go over the ring R = K[G0]. Furthermore, a recent paper [5] of the authors studies the general crossed product situation R*H with H finite and offers a rather complicated scheme for describing the prime ideals of R • H in terms of those of R. Since the situation here is certainly much more special, the general scheme, as expected, simplifies enormously and allows us to lift information from R = K[G0] to K[G]. Thus the proofs of the main results of this paper use crossed-product techniques, frequently well disguised, and ultimately rest on the structure of the primes in R = K[G0], that is, on the basic work of Roseblade. To describe these new results, we first require a number of definitions, most of which come from [8]. In the following definitions and theorems of the introduction, K[G] will always denote the group algebra of a polycyclic-by-finite group G over a field K.