Let $E$ be a nonsingular elliptic curve with ring of functions $k[E]$. Is $k[E]$ a unique factorization domain? I mean $E$ is a one-dimensional variety, so this should be right?
2026-03-25 21:50:06.1774475406
Is the ring of functions of an elliptic curve a UFD?
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The answer depends on whether you take the affine or projective version of your definition in the comments. For instance, the affine curve cut out by $y^2=x^3-x$ has a ring of global functions which is not a UFD, but it's projective completion given by $y^2z=x^3-xz^2$ has ring of global functions just $k$, so it's trivially a UFD.