Let $R$ be an integral domain, let $S \supset R$ be an extension of $R$ and let $x \in S$ be integral over $R$, i.e. there is a monic polynomial $P \in R[T]$ such that $P(x)=0$.
The kernel of the evaluation ring morphism $ev_x : R[T] \to R$ is a non-zero ideal $J$ of $R[T]$. Let $m$ be the minimum of the degrees of the non-zero polynomials in $J$. There must be a polynomial $P_1 \in J$ of degree $m$, but it might not be the only such polynomial (even if we add "up to a constant").
Whence my question : Is there an example of $R$ and $x \in S$ where there are two "non-associated" non-zero polynomials of minimal degree in $J$ ? In other words, I want $P_1, P_2 \in J$ of degree $m$ but there is no $c \in R$ such that $P_2=cP_1$.
Comments :
1) If $R[T]$ is a PIR (equivalently $R$ is field [I assumed that $R$ is a domain, so it can't be e.g. a product of fields]), then $P_1$ is unique and generates $J$.
2) Let $P_2 \in J$ be another polynomial of degree $m$. Then $P_1 - P_2$ might not be of smaller degree! Indeed, they might not be monic polynomials.
3) Maybe we could look at the extension of $J$ in $Frac(R)[T]$ (which is a PID)...
Let $k$ be a field, and $S=k[x]$ the polynomial ring in one indeterminate $x$ over $k$.
In the ring extension $R=k[x^2,x^3]\subset S=k[x]$ the element $x\in S$ is integral over $R$ but is killed by the two different minimal monic polynomials in $R[T]$: $$T^2-x^2,\quad T^2+x^2T-x^2-x^3 $$