Let $R$ be a D273: Division ring such that

(i)	$1_R$ is a D577: Multiplicative identity in $R$

Let $V$ and $W$ each be a D29: Vector space over $R$.
A D18: Map $f : V \to W$ is affine from $V$ to $W$ over $R$ if and only if \begin{equation} \forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \, x \in V^N : \forall \, r \in R^N \left[ \sum_{n = 1}^N r_n = 1_R \quad \implies \quad f \left( \sum_{n = 1}^N r_n x_n \right) = \sum_{n = 1}^N r_n f(x_n) \right] \end{equation}