\begin{definition}[Definition:Ring of Mappings/Additive Inverse]
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 additive inverse in the ring of mappings is defined by:
:$\forall f \in R^S : -f \in R^S : \forall s \in S : \map {\paren {-f} } x = -\map f x$
\end{definition}