Big projective modules over noetherian semilocal rings

We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^{\ast }(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of $V^{\ast }(R\mathrm{/}J(R))$, is isomorphic to the monoid of solutions in $(\No \cup{\infty})^k$ of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if $M$ is a submonoid of $(\No \cup{\infty})^k$containinganorderunit$(n_1,…, n_k)$of$\No^k$ which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as $V^{\ast }(R)$ for a noetherian semilocal ring such that $R\mathrm{/}J(R)\cong M_{n_1}(D_1)\times \mathrm{.}\mathrm{.}\mathrm{.}\times M_{n_k}(D_k)$ for suitable division rings $D_1,\mathrm{.}\mathrm{.}\mathrm{.},D_k$.