Let $U$ be a Noetherian UFD and let $D$ be a Noetherian integral domain (not known to be a UFD) such that $U \subseteq D$. Further assume that $U$ and $D$ have the same finite Krull dimension.
Of course, generally, an irreducible (=prime) element of $U$ may become reducible in $D$.
What can be said about such pairs of domains with the additional property that every irreducible element of $U$ remains irreducible in $D$?
An example: $U=\mathbb{C}[x^2]$, $D=\mathbb{C}[x^2][x^3]$; if I am not wrong, every irreducible element of $\mathbb{C}[x^2]$ remains irreducible in $D=\mathbb{C}[x^2][x^3]$ (though not prime).
Edit: If my above question is too general, then I wish to ask the following question: Given an irreducible element $u \in U$, can one find a "nice" criterion which guarantees that $u$ remains irreducible in $D$?
New edit: Another question:
If we further assume that $U \subseteq D$ is etale, then is it true that every irreducible element of $U$ remains irrdducible in $D$? or is it true that every prime element of $U$ remains prime in $D$?
Please see this recent question.
Thank you very much!