Definition:Ring of Mappings/Units

Definition
Let $\struct {R, +, \circ}$ be a ring with unity $1$. Let $S$ be a set.

Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$.

Let $f : S \to U_R$ is a mapping into the set of units $U_R$ of $R$.

From Unit of Ring of Mappings iff Image is Subset of Ring Units:
 * $f$ 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

 * Structure Induced by Ring Operations is Ring


 * Unit of Ring of Mappings iff Image is Subset of Ring Units