# Structure Induced by Commutative Ring Operations is Commutative Ring

## Theorem

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

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 commutative ring.

## Proof

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

From Structure Induced by Commutative Operation is Commutative, so is the pointwise operation $\circ$ induces on $R^S$.

The result follows by definition of commutative ring.

$\blacksquare$