\begin{definition}[Definition:Ring of Mappings/Pointwise Multiplication]
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 $R$.
<section notitle=true name=PointwiseMultiplication>
The pointwise operation $\circ'$ induced by $\circ$ on the ring of mappings from $S$ to $R$ is called '''pointwise multiplication''' and is defined as:
:$\forall f, g \in R^S: f \circ’ g \in R^S :$
:::$\forall s \in S : \map {\paren {f \circ’ g}} x = \map f x \circ \map g x$
</section>
\end{definition}