Correspondence theorem for ideals

Let \(R\) be a ring, \(I \subseteq R\) a two-sided ideal, and \(\phi\colon R \to R/I\) the quotient map.

1. The maps \(J \mapsto \bar{J} \coloneqq \phi(J) = J/I\) and \(\bar{J} \mapsto \inverseof{\phi}(\bar{J})\) are mutually inverse inclusion-preserving bijections \[ \begin{aligned} \correspond{\text{ideals } J \subseteq R \text{ with } J \contains I} &\mapstofrom \correspond{\text{ideals of } R/I} \\ J &\mapsto J/I \\ \inverseof{\phi}(\bar J) &\mapsfrom \bar{J}. \end{aligned} \] In particular, every ideal of \(R/I\) is of the form \(S/I\) for a unique ideal \(S \subseteq R\) containing \(I\).

2. The same maps give a bijection between subrings \(A \subseteq R\) containing \(I\) and subrings of \(R/I\).

3. (Third isomorphism theorem.) For ideals \(I \subseteq J \subseteq R\), there is a ring isomorphism \[ \frac{R/I}{J/I} \cong \frac{R}{J}. \]