Let $A = C^\omega(S^1)$ (resp. $C^\omega_{\mathbb C}(S^1)$) the ring of real-analytic real-valued (resp. complex valued) functions on the circle.
These rings have maximal ideals $\mathfrak m_p = \left \{ f \in A \, | \, f(p) = 0\right \}$ (for $p \in S^1$) and ideals $\mathfrak m_{p_1}^{e_1} \mathfrak m_{p_2}^{e_2} \cdots \mathfrak m_{p_n}^{e_n}$ (the ideal of functions having prescribed zeroes).
What I would like to prove is that there are no other ideals.
That would give a nice example of Dedekind rings: $C^\omega_{\mathbb C}(S^1)$ would be a PID (because it is not hard to give functions generating the aforementioned ideals) but $C^\omega(S^1)$ would be an example of Dedekind ring $A$ with $\mathrm{Cl}(A) = \mathbb Z/2$ (essentially because of the intermediate value theorem: only ideals $\mathfrak m_{p_1}^{e_1} \mathfrak m_{p_2}^{e_2} \cdots \mathfrak m_{p_n}^{e_n}$ with $e_1 + \cdots + e_n$ even are principal).
I feel like such a result, if true, must be classical, but I was unable to find references on those rings (unlike their algebraic counterpart: trigonometric polynomial rings $\mathbb R[S^1] = \mathbb R[X,Y]/(X^2+Y^2-1) \simeq \mathbb R[\cos \vartheta, \sin \vartheta]$ and $\mathbb C[S^1] = \mathbb C[X,Y]/(X^2+Y^2-1) \simeq \mathbb C[e^{\pm i \theta}]$).