Definition:Module of All Mappings

Definition
Let $\left({R, +_R, \times_R}\right)$ be a ring.

Let $\left({M, +_M, \circ}\right)_R$ 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 $+_M$
 * $\forall \lambda \in R: \forall f \in M^S: \forall x \in S: \left({\lambda \circ f}\right) \left({x}\right) = \lambda \circ f \left({x}\right)$.

Then $\left({M^S, +, \circ}\right)_R$ is the module of all mappings from $S$ to $M$.

In Module of All Mappings is Module, it is shown that $\left({M^S, +, \circ}\right)$ is an $R$-module.

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