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

Abstract

We consider the ring of real analytic functions defined on $[0,1]$, i.e. $$C^{\omega }[0,1]=\{f:[0,1]⟶\mathbb{R}|f\text{ is analytic on }[0,1]\}$$ 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: $$\text{For }\gamma \in [0,1],M_{\gamma }:=\{f\in C[0,1]|f(\gamma )=0\}$$ 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 $M_{\gamma }^{\omega }$ 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 $M_{\gamma }^{\omega }$ 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.