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My book shows that: $M$ is a maximal ideal of $R$ iff $R/M$ is a field. Let $M\subset J\subset R$.

They consider $R/J\cong (R/M)(J/M)$. They write $\overline R=R/M$ and $\overline J=J/M$. They claim that there is a one-to-one correspondence between the ideals $J\supset I$ of $R$ and $\overline J$ of $\overline R$. They also say that if we have an $R$-ideal $J$ that is a proper subset of $R$ (and $M$ is a proper subset of $J$), that the corresponding $\overline R$-ideal lies ‘properly’ between $\{\overline 0\}$ and $\overline R$, and vice versa. And then they conclude by saying: $M$ is a maximal ideal of $R$ $\iff$ $\{\overline 0 \}$ is a maximal ideal of $\overline R=R/M$.

So I can follow this properly, apart from their claim:

There is a one-to-one correspondence between the ideals $J\supset I$ of $R$ and $\overline J$ of $\overline R$.

Obviously there is a correspondence between $R$ quotiented by $J$, and $\overline R$ quotiented by $\overline J$, however, show can I lift this correspondence to a correspondence between the ideals $J$ and $\overline J$? I was given the hint to use the projection map $R \to R/I$, but I wouldn't know what to do with that. Could someone help me out?

Sha Vuklia
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The correspondences are these: \begin{alignat}{2} f\colon R&\longrightarrow&& R/I\\ \cup&&&\enspace\cup\\ J&\longmapsto&&\,\overline J=J/I\\ f^{-1}(\overline J)&\longleftarrow&&\enspace\overline J \end{alignat} As $f$ is surjective, you can easily checks these correspondence are inverse to each other.

Bernard
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