Assume $A$ is a commutative ring with unity and a domain. Let $A[k]$ be a generic non-trascendental extension of $A$. Let the element $k$ be such that, for a given polynomial $p(x)$ in $A[x]$, we have that $p(k)=0$ and for any other $g(x)$ such that $g(k)=0$, $g(x)$ must be a multiple of $p(x)$ in $A[x]$ (i.e. $p(x)$ is the minimal polynomial of $k$).
It seems to me that certain properties of $A[k]$ are solely based upon properties of the polynomial $p(x)$ and of the ring $A$ itself. But it seems like there are very few characterisations of this kind.
A question that arises is: given $A$ is Euclidean (or PID, UFD, Bèzout, GCD) for which polynomials $p(x)$ I have that $A[k]$ is Euclidean (or PID, UFD, Bèzout, GCD). Essentially, I am interested on when does the extension conserve a given property. I would also be interested to see if there are maybe some (non-trivial) necessary or sufficient conditions to have the inheritance of said property onto the extension ring.
There are some properties that are easy to characterize, for instance if $A$ is a field, then we have:
$$ p(x)\text{ is irreducible } \iff A[k]\text{ is a field }. $$
But I was wondering if there are more, maybe more complex characterisations.