Definition:Ring of Mappings/Pointwise Multiplication

Definition
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 $R$.

 The operation $\circ’$ induced by $\circ$ on the ring of mappings from $S$ to $R$ is called pointwise multiplication and is defined as:
 * $\forall f, g \in R^S: f \circ’ g \in R^S :$
 * $\forall s \in S : \map {\paren {f \circ’ g}} x = \map f x \circ \map g x$

Also see

 * Structure Induced by Ring Operations is Ring