Infinitely generated projective modules over pullbacks of rings

Abstract: We use pullbacks of rings to realize the submonoids $M$ of $(\N_0\cup{\infty})^k$ which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right $R$-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left $R$-modules is determined by the monoid $D(M)$ defined by reversing the inequalities determining the monoid $M$. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters \cite{FS}. We also provide a rich variety of examples of semilocal rings having non finitely generated projective modules that are finitely generated modulo the Jacobson radical.