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### Regularity of simple ring extensions, subrings and quotients [migrated]

Let $R$ be a regular UFD of zero characteristic, $I=(p)$ a prime ideal of $R$ and $Q(R)$ the field of fractions of $R$.
Assume $R[a]$ is integral and flat over $R$, for some $a \notin Q(R)$. Is it ...

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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 ...