# Questions tagged [unique-factorization-domains]

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### When UFD implies PID

The following result is too elementary, both to state and to prove, not to be known. Can someone give a reference? Is there any hope if you don't suppose UFD (i.e. move that from the hypothesis to ...
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### If R is UFD , then does R≅R[X,Y] imply R≅R[X]?

0 R is isomorphic to R[X,Y], but not to R[X] shows that it is possible to have commutative ring R with unity such that R≅R[X,Y] but R≆R[X]. My questions are: Is it possible to have an example of a ...
### If $R$ is UFD , then does $R \cong R[X,Y]$ imply $R \cong R[X]$?
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$ shows that it is possible to have commutative ring $R$ with unity such that $R \cong R[X,Y]$ but $R \ncong R[X]$. My questions are: Is it possible ...
### For which monic irreducible $f \in \mathbb{C}[x,y][T]$, $\mathbb{C}[x,y][T]/(f)$ is a UFD?
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic irreducible polynomial: $f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$, $a_j \in \mathbb{C}[x,y]$, $0 \leq j \leq n-1$. Denote \$B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}...