Abstract
We prove that for a noetherian semilocal ring R with exactly k isomorphism classes of simple right modules the monoid V*(R) of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of V*(R/J(R)), is isomorphic to the monoid of solutions in (ℕ0 ∪ {∞})k of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if M is a submonoid of (ℕ0 ∪ {∞})k containing an order unit (n1, . . . , nk) of 
which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as V*(R) for a noetherian semilocal ring such that R/J(R) ≅ Mn1(D1) × ⋯ × Mnk (Dk) for suitable division rings D1, . . . , Dk.
© Walter de Gruyter Berlin · New York 2010
