In 1905 Dickson observed that there are structures that have all the properties of a skewfield with the exception of one distributive property---these are nearfields. This gave rise to the study of the structure of distributive elements in a nearfield, or more generally, in a nearring. Here, given a finite Dickson nearfield $R$, the generalized distributive set defined as $D(\alpha, \beta)=\{ \gamma \in R \mid (\alpha +\beta )\circ \gamma=\alpha \circ \gamma +\beta \circ \gamma\}$, where $\circ$ is the multiplication of the Dickson nearfield, is studied and its structure is determined. An algorithm is derived to test whether $D(\alpha, \beta)$ is a subfield of $\Bbb{F}_{q^n}$. Moreover, sufficient conditions are obtained on $\alpha$ and $\beta$ for $D(\alpha, \beta)$ to be a subfield of $\Bbb{F}_{q^n}$. \par Furthermore, $R$-dimensions, $R$-basis, seed sets and seed number of $R$-subgroups of the Beidleman near-vector spaces $R^m$ are studied. \par The author gives an extensive background for the subject in the first two sections. The construction of a Dickson finite nearfield is explained and a lot of results characterizing the $R$-subgroups of $R^m$ are given.