\begin{definition}[Definition:Ring of Mappings/Unity]
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1$.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$.
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:
:$\quad \forall s \in S : \map {f_1} x = 1$
\end{definition}