Definition:Ring of Mappings

Definition
Let $\struct {R, +, \circ}$ be a ring.

Let $S$ be a set.

Let $R^S$ be the set of all mappings from $S$ to $R$.

The ring of mappings from $S$ to $R$ is the algebraic structure $\struct {R^S, +', \circ'}$ where $+'$ and $\circ'$ are the operations induced on $R^S$ by $+$ and $\circ$. The operation $+’$ is the pointwise addition and the operation $\circ’$ is the pointwise multiplication.

From Structure Induced by Ring Operations is Ring, $\struct {R^S, +', \circ'}$ is a ring.

Also denoted as
It is usual to use the same symbols for the induced operations on the ring of mappings from $S$ to $R$ as for the operations that induces them.

Also see

 * Structure Induced by Ring Operations is Ring


 * Structure Induced by Ring with Unity Operations is Ring with Unity


 * Structure Induced by Commutative Ring Operations is Commutative Ring


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