Distributive decomposition of near-vector spaces

TL;DR

This paper characterizes near-vector spaces by decomposing them into direct sums of vector spaces over division rings, providing new structural insights and dimension relations.

Contribution

It introduces two novel characterizations of near-vector spaces, linking their regularity to direct sum decompositions and dimension expressions.

Findings

Near-vector spaces can be expressed as direct sums over division rings.

Regularity of near-vector spaces is characterized by these decompositions.

Dimension of near-vector spaces relates to the dimensions of summands.

Abstract

This paper provides two characterizations of regularity for near-vector spaces: first, by expressing them as a direct sum of vector spaces over division rings formed by distributive elements; second, by expressing their dimension in term of the dimension of these summands. These results offer new insights into the structure and properties of near-vector spaces.