Questions about commutative rings, their ideals, and their modules.

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Factorization of ideals in a coordinate ring (Dedekind domain)

Consider $f \in \mathbb{C}[X,Y]$ an irreducible curve non singular. Let $A = \mathbb{C}[X,Y] / (f)$ be the coordinate ring of $f$ and choose a curve $g \in \mathbb{C}[X,Y]$ with no component in common ...