Let $I$ be an invertible ideal in an integral domain $R$. I claim that it is a maximal ideal. Please tell my i am correct or not.
Here is my attempt: If $I\subseteq J$, for some proper ideal $J$ of $R$. Then $R=II^{-1}\subseteq JI^{-1}$ and hence $JI^{-1}=R$. It implies that $J=I$.
The issue is that you seem to be assuming that $JI^{-1}\subseteq R$, which need not be the case.
Let's look at what happens with a specific example; the simplest thing we could possibly look at is $R=\Bbb Z$. Let's take $I=(4)$ and $J=(2)$; then $JI^{-1}=(\frac 12)$ (i.e. this is the additive subgroup of $\mathbb Q$ generated by the element $\frac12$), and clearly this is not actually contained in $\Bbb Z$.