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 |
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Additional Information: | Under review for Proceedings of the LMS. |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundations MSC 2010, the AMS's Mathematics Subject Classification > 06 Order, lattices, ordered algebraic structures MSC 2010, the AMS's Mathematics Subject Classification > 16 Associative rings and algebras MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations MSC 2010, the AMS's Mathematics Subject Classification > 55 Algebraic topology |

Depositing User: | Mr. Amit Kuber |

Date Deposited: | 18 Feb 2013 |

Last Modified: | 20 Oct 2017 14:13 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/1942 |

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