Colon ideal of fractional ideals is itself a fractional ideal

Question

I received this question on homework in my homological algebra class and I need some guidance.

Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of $R$ is an $R$-submodule of $K$ such that there is some nonzero $r \in R$ such that $rI \subset R$.

Let $(I:J) = \{k \in K \mid kJ \subset I\}$ (The colon ideal). I need to show that the colon ideal is also a fractional ideal.

I have shown that it is a $R$-submodule of $K$, but I cannot figure out how to show that there is some $r \in R$ such that $r(I:J) \subset R$.

What are some thoughts?

Answer 1

$aI\subseteq R$, $bJ\subseteq R$ with $a,b\in R-\{0\}$; take $r=abj$ for some $j\in J$, $j\ne 0$.