The subspace structure of finite dimensional Beidleman near-vector spaces

Abstract

The subspace structure of Beidleman near-vector spaces is investigated. We characterise finite dimensional Beidleman near-vector spaces and we classify the R-subgroups of finite dimensional Beidleman near-vector spaces. We provide an algorithm to compute the smallest R-subgroup containing a given set of vectors. Finally, we classify the subspaces of finite dimensional Beidleman near-vector spaces.

On the generalized distributive set of a finite nearfield

“…The material presented in this section is taken from [4], in which finite dimensional Beidleman near-vector spaces were characterized as follows:…”

“…Theorem 2.1. ( [4]) Let R be a (left) nearfield and M R a (right) nearring module. M R is a finite dimensional near-vector space if and only if M R ∼ = R m for some positive integer m = dim(M R ).…”

“…Later André [1] introduced another notion of near-vector spaces and used automorphisms in the construction. André's version has been studied in many papers, and theses, for example [10] while Beidleman near-vector spaces had not since his thesis until recently the authors of [4] added to the body of existing work.…”

“…In [4], the authors characterized the R-subgroups and subspaces of finite dimensional Beidleman near-vector spaces. In this paper we continue the work of that paper.…”

“…In subsection 1.1 we have used the results by Hull and Dobell ( [11]) to explain the finite Dickson construction. In section 2 we recall some results from [4] characterizing the R-subgroups of R m . We attempt to give sufficient background for the paper because of the nearfield experts that are new to finite dimensional Beidleman near-vector spaces.…”