\begin{definition}[Definition:Ring of Mappings/Units]
Let $\struct {R, +, \circ}$ be a ring with unity $1$.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$.
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:
::$f^{-1} \in R^S : \forall x \in S: \map {\paren {f^{-1} } } x = \map f x^{-1}$
\end{definition}