Grothendieck Rings of Theories of Modules

TL;DR

This paper computes the Grothendieck ring of modules over any unital ring, confirming a conjecture that it is non-trivial for non-zero modules, using homology theory of simplicial complexes.

Contribution

It provides a general computation of the Grothendieck ring for modules over any unital ring and proves a conjecture about its non-triviality.

Findings

Grothendieck ring of a module is non-trivial if the module is non-zero

Explicit computation of the Grothendieck ring for modules over any unital ring

Application of homology theory techniques in the proof

Abstract

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, $K_0(M_\mathcal R)$, of a right $R$-module $M$, where $\mathcal R$ is any unital ring. As a corollary we prove a conjecture of Prest that $K_0(M)$ is non-trivial, whenever $M$ is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.