Let G be a polycyclic-by-finite group and let K be a field. Then a well-known theorem of P. Hall [7, Corollary 10.24 asserts that the group algebra K[q is both a right and left Noetherian ring. In particular, if P is a prime ideal of K[C;I, then K[C;I/P is a prime Noetherian ring and work of Goldie [3, Theorem 1.371 implies that K[C;I/P has a classical right ring of quotients Z?(K[q/P) which is simple Artinian. In analogy with [I], we define the heart of P, S(P), to be the the center of this ring of quotients. Thus S(P) is a field containing K and the aim of this paper is to study the extension &‘(P)/K. We show first that if P is a particularly nice prime, a standard prime, then S(P) is equal to 2?(9’(K[q/P)), the ring of quotients of the center of K[C;I/P. Furthermore, the latter is a finite extension of the image of the center of K[Gj in K[G]/P. Now it follows from a recent important paper of Roseblade [8] that every prime P is closely related to a standard prime. Because of this, we can therefore show that for all such P, S(P) is a finitely generated extension of K. In particular, the transcendence degree of Z(P) over K is finite and we denote this number by c.r. P, the central rank of P. We also show that if N u G and if P is a prime ideal of K[G], with H(P) a nonabsolute field, then there exists a prime Q of K[N] with c.r. Q < c.r. P and P n K[NJ = flrEC p. Since the primitive ideals of K[G] are essentially those primes of central rank 0, we therefore obtain a number of corollaries concerning the latter ideals.