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I was looking for an integral domain which is not a UFD and I've thought to

$\mathbb{C}[x,y]/(x-y^2)$

My questions are:

Is it right?

Is there any easier example?

wood
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    That's isomorphic to $\mathbb{C}[x]$. You can take $\mathbb{C}[x, y]/(x^2 - y^3)$ although it takes a little effort to actually prove that this works. – Qiaochu Yuan Aug 25 '22 at 16:42
  • Ok, to be clear. I though to write $y^2$ both as $y \cdot y$ that as $x$ but in the quotient $x$ is not more irreducible. Is it right? Now, to prove your state, it is sufficient to prove that, in the quotient $x$ and $y$ are still irreducible, isn't it? – wood Aug 25 '22 at 16:48
  • https://math.stackexchange.com/questions/3016671/when-are-quadratic-rings-of-integers-unique-factorization-domains gives you a nice class of such rings. – Severin Schraven Aug 25 '22 at 17:48
  • DaRT query for commutative domains that aren't UFDs. – rschwieb Aug 25 '22 at 18:05

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