Non-monic minimal polynomial implies non-integral element?

Question

In a ring extension $R\subseteq S$, in order to check whether an element of $S$ is integral over $R$, I wonder why it is enough to check the minimal polynomial? In a concrete example, I found that the minimal polynomial was not monic (which we require for the element to be integral), how can I conclude from here that there is no monic polynomial with coefficients in $R$ annihilating the considered element of $S$? Couldn’t there be such a polynomial that simply has higher degree than the minimal one? Thanks for your help!