----- > [!proposition] Proposition. ([[the extension-contraction correspondence]]) > Let $A,B$ be [[ring|rings]] and $\varphi:A \to B$ a [[ring homomorphism]]. There is a [[bijection]] $\{ \text{contracted ideals of }A \} \leftrightarrow \{ \text{extended ideals of }B \}$ given by sending an [[ideal]] $\mathfrak{a}$ of $A$ to its [[extension of an ideal|extension]] $\mathfrak{a}^{e} \subset B$ and an [[ideal]] $\mathfrak{b}$ of $B$ to its [[contraction of an ideal|contraction]] $\mathfrak{b}^{c} \subset A$. ^proposition > [!proof]+ Proof. ([[the extension-contraction correspondence]]) > From the equivalences in the notes on extending and contracting ideals, it is immediate that the mentioned maps are mutual inverses. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```