Definition:Ring of Mappings

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Definition

Let $\struct {R, +, \circ}$ 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 algebraic structure $\struct {R^S, +', \circ'}$ where $+'$ and $\circ'$ are the operations induced on $R^S$ by $+$ and $\circ$.


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


Pointwise Addition

The operation $+’$ induced by $+$ on the ring of mappings from $S$ to $R$ is called pointwise addition and is defined as:

$\forall f, g \in R^S: f +’ g \in R^S :$
$\forall s \in S : \map {\paren {f +’ g}} x = \map f x + \map g x$


Pointwise Multiplication

The operation $\circ’$ induced by $\circ$ on the ring of mappings from $S$ to $R$ is called pointwise multiplication and is defined as:

$\forall f, g \in R^S: f \circ’ g \in R^S :$
$\forall s \in S : \map {\paren {f \circ’ g}} x = \map f x \circ \map g x$


Zero of Ring of Mappings

The zero of the ring of mappings is the constant mapping $f_0 : S \to R$ defined by:

$\quad \forall s \in S : \map {f_0} x = 0$

where $0$ is the zero in $R$


Additive Inverse

The additive inverse in the ring of mappings is defined by:

$\forall f \in R^S: -f \in R^S : \forall s \in S :\map {\paren {-f}} x = - \map f x$


Unity of Ring of Mappings

Let $\struct {R, +, \circ}$ be a ring with unity $1$.

From Structure Induced by Ring with Unity is a Ring with Unity, the ring of mappings from $S$ to $R$ is a ring with unity $f_1 : S \to R$ defined by:

$\quad \forall s \in S : \map {f_1} x = 1$


Units of Ring of Mappings with Unity

Let $\struct {R, +, \circ}$ be a ring with unity $1$.

Let $f : S \to U_R$ is a mapping into the set of units $U_R$ of $R$

From Unit of Ring of Mappings iff Image is Subset of Ring Units, $f$ is a unit in the ring of mappings from $S$ to $R$ and the inverse of $f$ is the mapping defined by:

$f^{-1} \in R^S : \forall x \in S: \map {\paren {f^{-1}}} x = \map f x^{-1}$


Commutativity of Ring of Mappings

Let $\struct {R, +, \circ}$ be a commutative ring.

From Structure Induced by Commutative Ring is a Commutative Ring, 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