\begin{definition}[Definition:Module of All Mappings]
Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {M, +_M, \circ}_R$ be an $R$-module.
Let $S$ be a set.
Let $M^S$ be the set of all mappings from $S$ to $M$.
Let:
:$+$ be the operation induced on $M^S$ by $+_M$
:$\forall \lambda \in R: \forall f \in M^S: \forall x \in S: \map {\paren {\lambda \circ f} } x = \lambda \circ \paren {\map f x}$
Then $\struct {M^S, +, \circ}_R$ is the '''module of all mappings from $S$ to $M$'''.
\end{definition}