Theorem 1. Let \(G\) be a virtually polycyclic group. Let \(U\) be a \(G\)-graded ring with a unit in each degree, such that \(U_ 1\) is Noetherian. Then the induction map \[ (1)\quad \oplus_{H\subset G\text{ finite}}K_ 0'U_ H\to K_ 0'U_ G \] is surjective, where \(U_ H\) is the part of \(U\) supported on \(H\), for each \(H\subset G\). -- \textit{S. Rosset} showed [Lect. Notes Math. 844, 35-45 (1981; Zbl 0462.16005)] when \(k\) is a field and \(G\) is a prime virtually polycyclic group, letting \(Q_ c(kG)\) denote the simple Artinian classical fraction ring of \(kG\), and writing \[ a=| H|_{H\subset G\text{ finite}} \] \(b=\text{length}(Q_ ckG)\), \(c=\) least common denominator \(\chi(M)\), \(M\) f.g. \(kG\)-module, that \(a| b\) and \(b| c\). Also he showed that whenever the induction map \(\oplus_{H\subset G\text{ finite}}K_ 0'(kH)\to K_ 0'(kG)\) is surjective, \(c| a\). However, Theorem 1 implies that this map is indeed surjective, so we obtain the solution of the Goldie rank conjecture: Theorem 2: length \(Q_ ckG=| H|_{H\subset G\text{ finite}}\).