One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every finitely generated ideal is principal$^1$, since it is a Bézout domain. However, it is not a UFD, for there are functions with infinitely many roots, and the only irreducible. Since every f.g. ideal is principal, every torsionfree $\mathcal O(D)$-module is flat (the converse always holds). Such rings are never noetherian, and I believe they have infinite Krull dimension. In some sense thus, they are not nicely behaved rings.

Lacking any knowledge or example, I am wondering if this has any relevancy. That is, are there any interesting applications of this fact to complex analysis (or any other related field), is this used in any fruitful manner in any relevant theorem, or is it just some curious consequence of

$1.$ See page 103, Theorem 5.20 here. Another very elegant proof due to Wedderburn can be found in Remmert's Classical Topics in Function Theory, Ch. 6, $\S 3$, 2 (page 137).


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