Let $G$ be a Lie group, $T:G\to GL(\mathbb{V})$ a representation of $G$ in a vector space $\mathbb{V}$. A $\mathbb{V}$-valued one-cocycle is a (smooth) map $S:G\to V$ satisfying the property $S(fg)=T(f)\big( S(g)\big) +S(f)$, for all $f,g\in G$. In the book I'm studying, the author says that: given a one-cocycle $S$, one defines the associated affine $G$-module as the space $\mathbb{V}\oplus \mathbb{R}$ acted upon by the following representation $\hat{T}$ of $G$: $$\hat{T}(f)\big( (v,\lambda )\big) =(T(f)\big( v\big) +\lambda S(f),\lambda )$$.
What is the definition of an affine $G$-module based on $\mathbb{V}$, and what does equivalence of two affine modules mean?