Fundamental theorem of geometry without the 1-to-1 assumption

It is proved that any mapping of an $n$-dimensional affine space over a division ring $\mathbb{D}$ onto itself which maps every line into a line is semi-affine, if $n\in \{2,3,\dots \}$ and $\mathbb{D}\ne \mathbb{Z}_{2}$. This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also - if the reader is interested in $\mathbb{D}$ other than $\mathbb{R}$, $\mathbb{Z}_{p}$, or $\mathbb{C}$ - division rings.