Structure Induced by Commutative Ring Operations is Commutative Ring
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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.
$\blacksquare$