Structure Induced by Ring Operations is Ring

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Theorem

Let $\left({R, +, \circ}\right)$ be a ring.

Let $S$ be a set.


Then $\left({R^S, +', \circ'}\right)$ is a ring, where $+'$ and $\circ'$ are the operations induced on $R^S$ by $+$ and $\circ$.


Proof

As $R$ is a ring, both $+$ and $\circ$ are closed on $R$ by definition.

From Closure of Pointwise Operation on Algebraic Structure, it follows that both $+'$ and $\circ'$ are closed on $R^S$:

$\forall f, g \in R^S: f +' g \in R^S$
$\forall f, g \in R^S: f \circ' g \in R^S$


By Structure Induced by Abelian Group Operation is Abelian Group, $\left({R^S, +'}\right)$ is an abelian group.

By Structure Induced by Associative Operation is Associative, $\left({R^S, \circ'}\right)$ is a semigroup.

From Pointwise Operation on Distributive Structure is Distributive, $\circ'$ is distributive over $+'$.

The result follows by definition of ring.

$\blacksquare$


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