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<p>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.&#xa0;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 <i>M</i>. Although in the case of near-vector spaces all elements of <i>M</i> 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).</p>

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Why is the Category of Near-Vector Spaces Abelian?

  • 期刊论文

  • 发表日期：

    2025年12月10日

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  • Zurab Janelidze,
  • Sophie Marques,
  • Daniella Moore

摘要

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).