Commutativity of Ring of Continuous Mappings

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Theorem

Let $\struct {S, \tau_{_S} }$ be a topological space.

Let $\struct {R, +, *, \tau_{_R} }$ be a commutative topological ring.

Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.

Then:

$\struct{\map C {S, R}, +, *}$ is a commutative ring

Proof

From Structure Induced by Commutative Operation is Commutative:

the ring of mappings $\struct{R^S, +, *}$ is commutative

From Ring of Continuous Mappings is Subring of All Mappings:

$\struct{\map C {S, R}, +, *}$ is a subring of $\struct{R^S, +, *}$

From Subring of Commutative Ring is Commutative:

$\struct{\map C {S, R}, +, *}$ is commutative

$\blacksquare$

Also see

• Ring of Continuous Mappings is Subring of All Mappings

• Zero of Ring of Continuous Mappings

• Additive Inverse in Ring of Continuous Mappings

• Unity of Ring of Continuous Mappings