# Definition:Module of All Mappings

## Definition

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 $M$.

## Examples

The most important case of this example is when $M = R$.