Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular.
Recall that a commutative noetherian ring is called regular if all its localizations at maximal ideals are regular local rings.
Let $w$ be algebraic over $A$ (I do not mind to assume that $w$ is integral over $A$. I do NOT want to assume that $w$ is in the field of fractions of $A$), and assume that $A[w]$ is noetherian.
My question: When is $A[w]$ regular? (I suspect there exists a counterexample, even under the additional conditions: $A$ is a UFD, $w$ is integral over $A$).
Notice the following question, which deals with $w$ transcendental over $A$: Does the regularity of $A$ imply the regularity of $A[X]$?