----- > [!proposition] Proposition. ([[module homomorphism is injective iff kernel is trivial iff is a monomorphism]]) > Let $R$ be a [[ring]]; $M,N$ $R$-[[linear map|modules]]. An $R$-[[linear map]] $\varphi:M \to N$ is a [[monomorphism]] if and only if its [[kernel of a module homomorphism|kernel]] is trivial, if and only if it is an [[injection]]. ^proposition > [!proof]- Proof. ([[module homomorphism is injective iff kernel is trivial iff is a monomorphism]]) > The proof in [[group homomorphism is injective iff kernel is trivial iff is a monomorphism]] translates from the [[category]] $\mathsf{Grp}$ of [[group|groups]] to the [[category]] $R$-$\mathsf{Mod}$. ----- #### ---- #### 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 > ```