Quotient of two ideals, considered as a fractional ideal

Question

Let $R$ be an integrally closed Noetherian domain and $K$ be its field of fractions. Consider two ideals $M, I \in R$ such that $I \subset M$ and $M$ is maximal ideal in $R$.

Is it true, that if a quotient $I:M$ considered as a quotient of two fractional ideals (and therefore a fractional ideal in itself) is unequal to $I$, then the same quotient considered as a quotient of two ideals from $R$ is also unequal to $I$?

Answer 1

Note that $R$ Noetherian and integrally closed implies that $R$ is completely integrally closed, i.e. $(J :_K J) = R$ for every ideal $J$. (This follows because when $k \in K$ is such that $kJ \subseteq J$ for a finitely generated ideal $J$, $k$ is integral over $R$).

Trivial Fact: Let $R$ be a completely integrally closed domain with quotient field $K$. Let $I \subseteq J$ ideals. Then $(I :_R J) = (I :_K J)$.

This settles your question affirmatively.