Structure Induced by Ring Operations is Ring

Theorem

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

Let $S$ be a set.

Then $\struct {R^S, +', \circ'}$ is a ring, where $+'$ and $\circ'$ are the pointwise 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, $\struct {R^S, +'}$ is an abelian group.

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

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

The result follows by definition of ring.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.10$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $8$