Ring Of Real Analytic Functions on $[0,1]$

TL;DR

This paper investigates the structure of ideals in the ring of real analytic functions on [0,1], proving it is a principal ideal domain with all maximal ideals generated by a single element, contrasting with the continuous case.

Contribution

It establishes that the ring of real analytic functions on [0,1] is a PID and characterizes its maximal ideals, which is a novel structural insight.

Findings

The maximal ideals in the ring are precisely the contractions of pointwise ideals from continuous functions.

Each maximal ideal in the ring is principal, generated by a single element.

The ring of real analytic functions on [0,1] is a principal ideal domain.

Abstract

We consider the ring of real analytic functions defined on $[0,1]$, i.e. $$C^{\omega}[0,1] =\lbrace f :[0,1] \longrightarrow \mathbb{R} | f \text{ is analytic on } [0,1]\rbrace$$ In this article, we explore the nature of ideals in this ring. It is well known that the ring $C[0,1]$ of real valued continuous functions on $[0,1]$ has precisely the following maximal ideals: $$\text{For } \gamma \in [0,1], M_{\gamma} := \lbrace f \in C[0,1] | f(\gamma) =0\rbrace$$ It has been proved that each such $M_{\gamma}$ is infinitely generated, in-fact uncountably generated. Observe that $C^{\omega}[0,1]$ is a subring of $C[0,1]$ We prove that for any $\gamma$ in $[0,1]$, the contraction $M^{\omega}_{\gamma}$ of $M_{\gamma}$ under the natural inclusion of $C^{\omega}[0,1]$ in $C[0,1]$ is again a maximal ideal (of $C^{\omega}[0,1]$ ), and these are precisely all the maximal ideals of $C^{\omega}[0,1]$. Next we prove that each $M^{\omega}_{\gamma}$ is principal (though $M_{\gamma}$ is uncountably generated). Surprisingly, this forces all the ideals of the ring $C^{\omega}[0,1]$ to be singly generated, i.e. $C^{\omega}[0,1]$ is a PID.