Definition:Module of All Mappings

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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 $M = R$.

Module on Cartesian Product

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $n \in \N_{>0}$.

Let $+: R^n \times R^n \to R^n$ be defined as:

$\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_R \beta_1, \ldots, \alpha_n +_R \beta_n}$

Let $\times: R \times R^n \to R^n$ be defined as:

$\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_R \alpha_1, \ldots, \lambda \times_R \alpha_n}$


Then $\struct {R^n, +, \times}_R$ is the $R$-module $R^n$.


Also see


Sources

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