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### What can be said about a regular quotient (by a principal prime ideal) of a polynomial ring? [migrated]

Let $f \in \mathbb{C}[x_1,\ldots,x_n]$ be irreducible (so (f) is a prime ideal).
Assume $S:=\mathbb{C}[x_1,\ldots,x_n]/(f)$ is regular, where, by definition, a noetherian ring is regular is all its ...

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### Existence of Factor rings of UFDs which are UFDs

Suppose that $X=Spec(A)$ is an affine variety over an algebraically closed field $k$ which is normal and such that $Cl(X)=0$.
I am interested in hypersurfaces of $X$ which again satisfy this ...

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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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### First-order UFD (factorial ring) condition / pre-Schreier rings

All rings in this post are commutative and with $1$.
Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...