Correspondence theorem for rings

Statement

Let $R$ be a ring, let $I$ be a two-sided ideal of $R$ and let $\pi \colon R \to R/I$ be the quotient map. Let $\mathcal{A}$ be the set of the left ideals $J$ of $R$ with $I \subseteq J$. Let $\mathcal{B}$ be the set of the left ideals of $R/I$. The map $\mathcal{A} \to \mathcal{B}$, $J \mapsto J/I = \pi(J)$, is a bijection. Its inverse is $K \mapsto \pi^{-1}(K)$. This bijection and its inverse preserve inclusion.

Remarks

The same bijection holds for right ideals, for two-sided ideals, and for subrings that contain $I$. For two-sided ideals $J \supseteq I$, the third isomorphism theorem gives $(R/I)/(J/I) \cong R/J$. The ideals of $\mathbb{Z}/12\mathbb{Z}$ correspond to the ideals $d\mathbb{Z}$ with $d \mid 12$, so there are six of them.

Explanations

L1 human

The ideals of a quotient ring $R/I$ are the ideals of $R$ that contain $I$, seen in the quotient. So you can understand the ideals of $R/I$ from the ideals of $R$. For example, the ideals of $\mathbb{Z}/12\mathbb{Z}$ come from the divisors $1, 2, 3, 4, 6, 12$ of $12$.