I am having a confusion regarding the correspondence theorem for rings.I want to know if the following is correct.
Let $R$ be a ring and $I$ be an ideal of $R$.Consider the sets $\mathcal G$ and $\mathcal N$to be respectively the set of all ideals of $R$ containing $I$ and the set of all ideals of $R/I$.Then for each ideal $J$ in $R$ containing $I$,$J/I$ is an ideal of $R/I$ i.e. $J\in \mathcal G\implies J/I\in \mathcal N$.Actually the quotient map $q:R\to R/I$ gives rise to a map $\overline{q}:\mathcal{G\to N}$ .Now for each $K\in \mathcal N$,there exists $J\in \mathcal G$ such that $K=J/I$ which implies that $K=\overline q(J)$ for some $J$,that is the map $\overline q$ is surjective.Now we can also say that the map $\overline{q}$ is injecctive by showing that $J_1/I=J_2/I\implies J_1=J_2$ for $J_1,J_2\in \mathcal G$.Thus we have a bijection between $\mathcal G$ and $\mathcal N$.
Am I right?Please correct me if I have made any mistake.