Infinitely generated projective modules over pullbacks of rings

TL;DR

This paper constructs specific semilocal rings using pullbacks to realize certain monoids as classes of countably generated projective modules, revealing asymmetries between left and right modules and providing new examples of rings with unusual projective module properties.

Contribution

It introduces a method to realize particular monoids as classes of projective modules over semilocal rings using pullbacks, and explores the asymmetry between left and right projective modules.

Findings

Existence of semilocal rings with all right modules free but not left modules.

Construction of rings with non-finitely generated projective modules that are finitely generated modulo the Jacobson radical.

Identification of monoids corresponding to classes of projective modules via linear inequalities.

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.