Zero of Ring of Bounded Continuous Real-Valued Functions

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Theorem

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

Let $\R$ denote the real number line.

Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.

Then:

the zero of $\struct{\map {C^*} {S, \R}, +, *}$ is the constant mapping $0_{\R^S} : S \to \R$ defined by:

$\forall s \in S : \map {0_{\R^S}} s = 0$

Proof

Let $\struct{\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.

From Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions:

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

From Zero of Ring of Continuous Real-Valued Functions:

the zero of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $0_{\R^S} : S \to R$ defined by:

$\forall s \in S : \map {0_{\R^S}} s = 0$

From Zero of Subring is Zero of Ring:

$0_{\R^S}$ is the zero of $\struct{\map {C^*} {S, \R}, +, *}$

$\blacksquare$

Also see

• Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions

• Additive Inverse in Ring of Bounded Continuous Real-Valued Functions

• Unity of Ring of Bounded Continuous Real-Valued Functions

• Ring of Bounded Continuous Real-Valued Functions is Commutative

Sources

1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.4$