The fraction ideal is the ideal $$(I:J)=\{x\in R:xJ\subseteq I\}$$where $I, J$ are ideals in $R$.
My question is, why is it called a fraction ideal? (Sometimes it is called the "Ideal quotient of $I$ by $J$".) After learning about the isomorphism theorems, I kind of see why the quotient ring is called that. Correct me if I am wrong, but we are, in a way, dividing by the ideal, taking representatives from the ring.
I understand how the fraction ideal is defined, but I do not see how this fraction ideal is analogous to a fraction.