# Two questions concerning integral dependence

Proposition 2.4 in Janusz's Algebraic Number Fields states that if $R$ is an integral domain with quotient field $K$, $L/K$ a field extension and $b \in L$ algebraic over $K$ with minimal polynomial $f \in K[X]$ and also integral over $R$, then all the coefficients of $f$ are integral over $R$ as well.

The proof given there is the following: Without loss of generality $f$ splits over $K$. Let $b_1,\dots,b_n$ be its roots in $L$. Let $g(X) \in R[X]$ be an equation of integral dependence of $b$ over $R$. Then $f$ divides $g$ in $K[X]$, as $g(b) = 0$, so $g(b_i) = 0$ for all $i$, i.e. all the $b_i$ are integral over $R$ as well. But $f(X) = (X - b_1)\cdots(X - b_n)$, so its coefficients are polynomials in the $b_i$ with coefficients in $R$ and we are done.

This proposition is useful, for example it implies that if $R$ is integrally closed, an element of $L$ is integral over $R$ if and only if it is algebraic over $K$ and its minimal polynomial has coefficients in $R$.

I have to questions, so let me describe the situation I am interested in.

Suppose $R$ is a ring (commutative and with unit), $S \subset R$ the multiplicatively closed subset of non-zero-divisors. Let $R \subset R'$ be a ring extension, such that all elements of $S$ are also non-zero-divisors in $R'$ and let $S'$ be the set of non-zero-divisors of $R'$. Then we get an extension of rings $R_S \subset R'_{S'}$, where $R_S$ and $R'_{S'}$ denote the respective localizations. Let $b \in R'_{S'}$ be integral over $R_S$ satisfying an irreducible $f(X) \in R_S[X]$ and assume that $b$ is also integral over $R$ satisfying $g(X) \in R[X]$.

Question 1 In the situation described above, is it true, that the coefficients of $f$ are integral over $R$?

Question 2 Is it true that the polynomial $g$, thought of as a polynomial living in $R_S[X]$, is divisible by $f$?

My feeling is that the answer to both questions is negative, as the conclusion $f|g$ in the proof of the original statement is due to the fact that $K[X]$ is a PID, and I know that $R[X]$ is a PID if and only if $R$ is a field. However, I could not come up with any counterexamples.

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I'm not clear what you are asking. Are you asking whether $b\in R'_{S'}$ integral over $R_S$ implies that it is integral over $R$? Or are you assuming that $b$ is integral over $R$ and over $R_S$? Also, without assumptions on $f(x)$, you cannot guarantee that $f$ divides $g$, since you can replace $f(x)$ with any monic multiple of itself. Are you implicitly assuming that $f$ and $g$ are irreducible in the corresponding rings of integers? – Arturo Magidin Jul 11 '12 at 18:40
I edited my question. Yes, $b$ is assumed to be integral over both $R$ and $R_S$, and $f$ should be irreducible. – Lennart Jul 11 '12 at 19:35