2009/03/17 by Herbera, Dolors, Prihoda, Pavel
#16D40 #16L30 #16P40 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.0903.2965
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 (\No ∪\∞\)k of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if M is a submonoid of (\No ∪\∞\)k containing an order unit (n1,..., nk) of \Nok 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.