Structure Induced by Ring with Unity Operations is Ring with Unity

Theorem

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.

Let $S$ be a set.

Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$.

Then $\struct {R^S, +', \circ'}$ is a ring with unity whose unity is $f_{1_R}: S \to R$, defined by:

$\forall s \in S: \map {f_{1_R} } s = 1_R$

Proof

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

We have from Induced Structure Identity that the constant mapping $f_{1_R}: S \to R$ defined as:

$\forall x \in S: \map {f_{1_R} } x = 1_R$

is the identity for $\struct {R^S, \circ'}$.

The result follows by definition of ring with unity and unity of ring.

$\blacksquare$