Why is the category of near-vector spaces abelian?

TL;DR

This paper provides a unified proof that both module categories over rings and near-vector spaces over scalar systems are abelian, by viewing them as subcategories of modules over a multiplicative monoid.

Contribution

It introduces a unified approach to prove the abelian nature of categories of modules and near-vector spaces using monoid modules, extending known results.

Findings

Both categories are abelian when viewed as subcategories of monoid modules.

Near-vector spaces form abelian categories despite the non-invertibility of some elements.

The approach generalizes the conditions under which these categories are abelian.

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. Andr\'e, 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).