# Definition:Ring of Mappings

## 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 (pointwise) 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 pointwise 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 pointwise 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 whose unity is $1$.

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

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

### 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 Operations is 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.