nLab semilinear map

A semilinear map (older term: semilinear transformation) from a left vector space $V$ over a division ring $k$ to a left vector space $W$ over a division ring $k'$ is an additive map $f: V\to W$ of the abelian groups of vectors together with a homomorphism of rings $\alpha : k\to k'$ such that the following weaker form of homogeneity holds:

$$f (\lambda v) = \alpha(\lambda) f(v), \,\,\,\,\,a\in k, v\in V$$

This notion is very much used in the classical affine and projective geometry over division rings, for example in the standard definition of a homothety.