\begin{definition}[Definition:Ring of Mappings/Zero]
Let $\struct {R, +, \circ}$ be a ring.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$.
From Structure Induced by Ring Operations is Ring, $\struct {R^S, +', \circ'}$ is a ring.
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 $R$
\end{definition}