----- > [!proposition] Proposition. ([[kernel iff ideal]]) > A subset of a [[ring]] $R$ is an [[ideal]] if and only if it is the [[kernel of a ring homomorphism]]. ^proposition > [!proof]- Proof. ([[kernel iff ideal]]) > **Kernel $\implies$ ideal.** We already know the [[kernel of a ring homomorphism|kernel]] is a [[subgroup]], so it is just absorption properties in the [[ideal]] definition which must be verified. Let $a \in \ker \varphi$ and $r \in R$. Then $\varphi(ar)=\varphi(a)\varphi(r)=0 \cdot \varphi(r)=0$ and likewise $\varphi(ra)=\varphi(r)\varphi(a)=\varphi(r) \cdot 0 =0$. Hence $ar$ and $ra$ live in $\ker \varphi$. > **Ideal $\implies$ Kernel.** On the other hand, any [[ideal]] $I$ is the [[kernel of a ring homomorphism]]: namely, the projection $\pi: R \to R / I$ (merely because it is the [[group homomorphism]] [[kernel of a group homomorphism|kernel]]). ----- #### ---- #### 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 > ```