Definition:Ring of Mappings

Definition
Let $\struct {R, +, \cdot}$ be a ring.

Let $S$ be a set.

Let $R^S$ be the set of all mappings from $S$ to $R$.

The ring of mappings from $S$ to $R$ is the ring $\struct {R^S, +', \circ'}$ induced on $R^S$ by $+$ and $\circ$.

That is, the ring of mappings is the set of mappings from $S$ to $R$ with pointwise addition and pointwise product.

From Structure Induced by Ring Operations is Ring, $\struct {R^S, +', \circ'}$ is a ring.

The ring operations on the ring of mappings from $S$ to $R$ are defined:
 * $\forall f, g \in R^S: f +’ g \in R^S : \forall s \in S : \map {\paren {f +’ g}} x = \map f x + \map g x$
 * $\forall f, g \in R^S: f \circ’ g \in R^S : \forall s \in S : \map {\paren {f \circ’ g}} x = \map f x \circ \map g x$

The zero of the ring of mappings is the constant mapping $f_0 : S \to R$, where:
 * $\quad 0$ is the zero in $R$
 * $\quad \forall s \in S : \map {f_0} x = 0$

The additive inverse in the ring of mappings is defined by:
 * $\forall f \in R^S: -f \in R^S : \forall s \in S :\map {\paren {-f}} x = - \map f x$

From Structure Induced by Ring with Unity is a Ring with Unity, if $R$ is a ring with unity then the ring of mappings from $S$ to $R$ is a ring with unity; namely the constant mapping $f_1 : S \to R$, where:
 * $\quad 1$ is the unity in $R$
 * $\quad \forall s \in S : \map {f_1} x = 1$

From Leigh.Samphier/Sandbox/Induced Structure Unit, if $R$ is a ring with unity and $f : S \to U_R$ is a mapping into the set of units $U_R$ of $R$ then $f$ is a unit in the ring of mappings from $S$ to $R$ and the inverse of $f$ is the mapping defined by:
 * $f^{-1} \in R^S : \forall x \in S: \map {\paren {f^{-1}}} x = \map f x^{-1}$

From Structure Induced by Commutative Ring is a Commutative Ring, if $R$ is a commutative ring then the ring of mappings from $S$ to $R$ is a commutative ring.

Also denoted as
It is usual to use the same symbols for the induced operations on the ring of mappings from $S$ to $R$ as for the operations that induces them.

Also see

 * Structure Induced by Ring Operations is Ring


 * Structure Induced by Ring with Unity Operations is Ring with Unity


 * Structure Induced by Commutative Ring Operations is Commutative Ring


 * Leigh.Samphier/Sandbox/Induced Structure Unit