Definition:Ring of Mappings/Additive Inverse

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 $R$.

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:
 * $\forall f \in R^S: -f \in R^S : \forall s \in S :\map {\paren {-f}} x = - \map f x$

Also see

 * Structure Induced by Ring Operations is Ring