Unit of Ring of Mappings iff Image is Subset of Ring Units/Unit of Ring of Mappings implies Image is Subset of Ring Units

Theorem

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

Let $U_R$ be the set of units in $R$.

Let $S$ be a set.

Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$.

Let $f \in R^S$ be a unit of $R^S$.

Then:

$\Img f \subseteq U_R$

where $\Img f$ is the image of $f$.

In which case, the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by:

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

$\forall x \in S: \map {f_{1_R} } x = 1_R$

Let $f^{-1}$ be the product inverse of $f$.

Let $x \in R$.

Then:

$\ds 1_R$

$=$

$\ds \map {f_{1_R} } x$

$\ds $

$=$

$\ds \map {\paren {f \circ' f^{-1} } } x$

$\ds $

$=$

$\ds \map f x \circ \map {f^{-1} } x$

Definition of Pointwise Operation $\circ'$

Similarly, $1_R = \map {f^{-1} } x \circ \map f x$

Hence $\map {f^{-1}} x$ is the product inverse of $\map f x$.

So $\map f x$ is a unit in $R$ and $\map f x^{-1} = \map {f^{-1} } x$.

The result follows.

$\blacksquare$