If $\alpha\notin R$ but $\alpha$ is the root of a polynomial in $R[x]$ do we study the “minimal polynomial” over $R$ for that root? (Would we still want it to be with leading coefficient 1? irreducibility uniqueness and minimal degree would be kept I guess)
For example if you consider $2/3\notin\Bbb Z$, the only polynomial over that comes to mind is $3x-2$
In general, if $R$ is a commutative ring, $S$ is an $R$-algebra, and $\alpha\in S$, then the analogue of the "minimal polynomial" of $\alpha$ is the kernel $I$ of the homomorphism of $R$-algebras $R[x]\to S$ sending $x$ to $\alpha$. When $R$ is a field, $R[x]$ is a PID, so this kernel $I$ is always a principal ideal and "the minimal polynomial" refers to a generator of the ideal. This makes sense more generally when $I$ is principal. So, for instance, when $R=\mathbb{Z}$, $S=\mathbb{Q}$, and $\alpha=2/3$, it is reasonable to say that $3x-2$ is the minimal polynomial since it generates the kernel of the homomorphism $\mathbb{Z}[x]\to\mathbb{Q}$ sending $x$ to $2/3$.
In general, though, the kernel may not be principal and there is no single polynomial that can be called the minimal polynomial.