# Grothendieck Rings of Theories of Modules

Kuber, Amit (2013) Grothendieck Rings of Theories of Modules. [MIMS Preprint]

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## 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.

Item Type: MIMS Preprint Under review for Proceedings of the LMS. MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundationsMSC 2010, the AMS's Mathematics Subject Classification > 06 Order, lattices, ordered algebraic structuresMSC 2010, the AMS's Mathematics Subject Classification > 16 Associative rings and algebrasMSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizationsMSC 2010, the AMS's Mathematics Subject Classification > 55 Algebraic topology Mr. Amit Kuber 18 Feb 2013 20 Oct 2017 14:13 http://eprints.maths.manchester.ac.uk/id/eprint/1942

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