  For any nearfield $(R,+, \circ)$, denote by $D(R)$ the set of all
distributive elements of $R$. Let $R$ be a finite Dickson nearfield that arises
from Dickson pair $(q,n)$. For a given pair $(\alpha, \beta) \in R^2$ we study
the generalized distributive set $ D(\alpha, \beta)$ where $"\circ"$ is the
multiplication of the Dickson nearfield. We find that $ D(\alpha, \beta)$ is
not in general a subfield of the finite field $\mathbb{F}_{q^n}$. In contrast
to the situation for $D(R)$, we also find that $D(\alpha, \beta)$ is not in
general a subnearfield of $R$. We obtain sufficient conditions on $\alpha,
\beta$ for $ D(\alpha, \beta)$ to be a subfield of $\mathbb{F}_{q^n}$ and
derive an algorithm that tests if $D(\alpha, \beta)$ is a subfield of
$\mathbb{F}_{q^n}$ or not. We also study the notions of $R$-dimension,
$R$-basis, seed sets and seed number of $R$-subgroups of the Beidleman
near-vector spaces $R^m$ where $m$ is a positive integer. Finally we determine
the maximal $R$-dimension of $gen(v_1,v_2)$ for $v_1,v_2 \in R^m$, where
$gen(v_1,v_2)$ is the smallest $R$-subgroup containing the vectors $v_1$ and
$v_2$.
