Definition:Ring of Mappings/Units

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.

From Leigh.Samphier/Sandbox/Unit of Ring of Mappings iff Image is Subset of Ring Units, if $R$ is a ring with unity and $f : S \to U_R$ is a mapping into the set of units $U_R$ of $R$ then $f$ is a unit in the ring of mappings from $S$ to $R$ and the inverse of $f$ is the mapping defined by:
 * $f^{-1} \in R^S : \forall x \in S: \map {\paren {f^{-1}}} x = \map f x^{-1}$

Also see

 * Leigh.Samphier/Sandbox/Unit of Ring of Mappings iff Image is Subset of Ring Units