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

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

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