---- > [!definition] Definition. ([[homomorphism of local rings]]) > Let $A$ and $B$ be [[local ring|local]] [[ring|rings]]; denote by $\mathfrak{m}_{A}$ and $\mathfrak{m}_{B}$ their respective unique [[maximal ideal|maximal]] [[ideal|ideals]]. Let $\varphi:A \to B$ be a [[ring homomorphism]]. > [[homomorphism of local rings#^justification|It is always true]] that $\varphi ^{-1}(\mathfrak{m}_{B}) \subset \mathfrak{m}_{A}$. We say that $\varphi$ is **homomorphism of local rings**[^1] if in fact the reverse inclusion holds: $\varphi ^{-1}(\mathfrak{m}_{B})=\mathfrak{m}_{A}.$ ^definition > [!justification] Why is it always true that $\varphi ^{-1}(\mathfrak{m}_{B}) \subset \mathfrak{m}_{A}$ ? > Let $a \in \varphi ^{-1}(\mathfrak{m}_{B})$, so that $\varphi(a) \in \mathfrak{m}_{B}$. If $a \notin \mathfrak{m}_{A}$, then[^2] $a$ is a [[unit]], [[ring homomorphisms preserve structure|and hence]] $\varphi(a)$ is a [[unit]]. Thus $\varphi(a) \notin \mathfrak{m}_B$ (since otherwise $\mathfrak{m}_{B}=(1)$). It follows that $a \in \mathfrak{m}_{A}$. > ^justification - [ ] Examples (page 46-47) - [ ] $(\text{Spec }A, \mathcal{O}_{\text{Spec } A})$, of course ---- #### [^1]: Or $\varphi$ may be called a **local ring homomorphism**. [^2]: If $a$ were not a unit, then $\langle \mathfrak{m}_{A} \cup \{ a \} \rangle$ would be a proper ideal containing $\mathfrak{m}_{A}$, hence equalling $\mathfrak{m}_{A}$, implying $a \in \mathfrak{m}_A$. ---- #### 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 > ```