Definition:Ring of Mappings/Zero

Note
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 $f_0 : S \to R$, where:
 * $\quad 0$ is the zero in $R$
 * $\quad \forall s \in S : \map {f_0} x = 0$

Also see

 * Structure Induced by Ring Operations is Ring