Why is the category of near-vector spaces abelian?

2025/01/28 by Zurab Janelidze, Janelidze, Zurab, Sophie Marques +3 · 1 citation
Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2501.16897

openalex publication_date 2025/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we present a unified proof of the fact that the category of modules over a ring and the category of near-vector spaces in the sense of J. André, over an appropriate scalar system (a 'scalar group'), are both abelian categories. The unification is possible by viewing each of these categories as subcategories of the (abelian) category of modules over a multiplicative monoid M. Although in the case of near-vector spaces all elements of M except one (the 'zero' element) are invertible, we show that this requirement is not necessary for the corresponding category to be abelian in analogy to the well-known fact that modules over a ring form an abelian category even if the ring is not a field (i.e., modules over it are not vector spaces).

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• Inner products for strongly regular near-vector spaces and duality for finite dimensional near-vector spaces

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