Intuition for ideal quotient / colon ideal?

I'm in an introductory commutative algebra class and have come across the notion of the ideal quotient $(I : J) = \{r \in R : rJ \subseteq I\}$ (where $I$ and $J$ are ideals of the commutative ring $R$).

I can manipulate these just fine, and I've seen their usefulness when it comes to working with primary decompositions, but I have no good intuition for what $(I:J)$ "really is". Why is this construction called a "quotient", and when are they used?

Solution 1:

In a Dedekind domain, for instance the ring of integers of a number field, then if $I$ and $J$ are ideals with $I\subseteq J$ then $I=J(I:J)$. In this case $(I:J)$ really is a quotient in the multiplicative sense, an ideal $X$ solving the equation $XJ=I$.