Why is the category of near-vector spaces abelian?
Zurab Janelidze, Sophie Marques, Daniella Moore
TL;DR
The paper develops a unified framework showing that both modules over a ring and near-vector spaces over a scalar group form abelian categories by embedding them into the abelian category of modules over a monoid $M$ ($M$-Mod). It introduces André modules over a multi-near-ring $(M,\mathbf{N})$ to capture the near-vector space structure and proves that this subcategory is abelian, with precise correspondences to ordinary modules and to near-vector spaces when specialized. A key contribution is a direct algebraic proof of the subspace hypothesis for near-vector spaces, establishing that submodules inherit near-vector space structure via a crucial closure equality. Collectively, these results connect near-vector space theory with classical module theory, clarifying when substructure closures preserve abelian properties and providing a robust categorical framework via multi-near-rings.
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).