----- > [!proposition] Proposition. ([[kernel iff submodule]]) > A subset of a [[module]] is a [[submodule]] if and only if it is the [[kernel of a module homomorphism]]. ^proposition > [!proof]- Proof. ([[kernel iff submodule]]) > ~ > **kernel $\implies$ submodule.** We already know $\ker \varphi$ [[kernel iff normal subgroup|is a]] [[subgroup]] of the [[abelian group|(abelian)]] [[group]] $M$. Now let $m \in \ker \varphi$, $r \in R$. Then $\varphi(rm)=r \varphi(m)=r0=0$, hence $rm \in \ker \varphi$. > **submodule $\implies$ kernel.** On the other hand, a [[submodule]] $N$ is the [[kernel of a module homomorphism]]: namely, of the canonical projection $\pi: M \to M / N$. ----- #### ---- #### 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 > ```