Linear sets in the projective line over the endomorphism ring of a finite field

Let $\mathrm{P}\mathrm{G}(1,E)$ be the projective line over the endomorphism ring $E=En{d}_q({\mathbb{F}}_{q^t})$ of the ${\mathbb{F}}_q$-vector space ${\mathbb F}{q^t}$\mathrm{.}Asiswellknownthereisabijection$\Psi:$\mathrm{P}\mathrm{G}$(1,E)\rightarrow{\cal G}{2t,t,q}$ with the Grassmannian of the $(t-1)$-subspaces in $\mathrm{P}\mathrm{G}(2t-1,q)$. In this paper along with any ${\mathbb{F}}_q$-linear set $L$ of rank $t$ in $\mathrm{P}\mathrm{G}(1,q^t)$, determined by a $(t-1)$-dimensional subspace $T^{\mathrm{\Psi }}$ of $\mathrm{P}\mathrm{G}(2t-1,q)$, a subset $L_T$ of $\mathrm{P}\mathrm{G}(1,E)$ is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring $E$. In particular the attention is focused on the relationship between $L_T$ and the set $L_T^{\mathrm{\prime }}$, corresponding via $\mathrm{\Psi }$ to a collection of pairwise skew $(t-1)$-dimensional subspaces, with $T\in L_T^{\mathrm{\prime }}$, each of which determine $L$. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set $L$ related to $T\in \mathrm{P}\mathrm{G}(1,E)$ is of pseudoregulus type if and only if there exists a projectivity $\phi$ of $\mathrm{P}\mathrm{G}(1,E)$ such that $L_T^{\phi }=L_T^{\mathrm{\prime }}$.