Equivalent definitions of fractional ideals

Question

Let $R$ be an integral domain and $K$ its field of fractions. The usual definition of fractional ideal $I$ ($I$ is an $R$-submodule of $K$) is that for some nonzero $r\in R$ we have $rI\subset R$, and the intuition is that $r$ clears the denominators in $I$. But I came across another definition: there are units $z$, $u\in K$ such that $zR\subset I\subset uR$. What is the relationship between these two definitions? Which one is "normally" used? What is the intuition for the second case?

Answer 1

Let $I$ be a fractional ideal according to definition $1$ which I believe includes also $I\neq(0)$.

Pick any $0\neq z\in I$ then $z\in K^\times$ and $zR\subset I$. Also, from $rI\subset R$ you get $I\subset\frac1rR$ and $u=\frac1r\in K^\times$.

If, on the other hand, you start from definition $2$, from $zR\subset I$ you get $z\cdot1\in I$ so that $I\neq(0)$. Next, write $u=\frac ab$ with $a$ and $b\in R$. Thus $I\subset uR=\frac abR$ implies $bI\subset aR\subset R$ which is the requirement of definition $1$ with $r=b$.