----- > [!proposition] Proposition. ([[characterization of left and right inverses in R-mod]]) > Let $R$ be a [[ring]] and $\varphi: M\to N$ an $R$-[[module]] [[linear map|homomorphism]]. Then: > $\varphi$ has a [[inverse map|left-inverse]] if and only if the [[chain complex of modules|sequence]] $0 \xrightarrow{} M \xrightarrow{\varphi} N \xrightarrow{} \text{coker }\varphi \to 0 $ is [[split short exact sequence of modules|split]] [[short exact sequence|(short)]] [[exact sequence|exact]]. > $\varphi$ has a [[inverse map|right-inverse]] if and only if the [[chain complex of modules|sequence]] $0 \xrightarrow{} \ker \varphi \xrightarrow{} M \xrightarrow{\varphi} N \xrightarrow{} 0$ is [[split short exact sequence of modules|split]] [[short exact sequence|(short)]] [[exact sequence|exact]]. ^proposition > [!justification] Motivation. > In the [[category]] $\mathsf{Set}$, we have the following: > > ![[characterization of injectivity and surjectivity in Set#^proposition]] > > In $R$-$\mathsf{Mod}$, although [[module homomorphism is injective iff kernel is trivial iff is a monomorphism|it is true that]] [[injection]] $\iff$ [[monomorphism]] [[module homomorphism is surjective iff cokernel is trivial iff is an epimorphism|and]] [[surjection]] $\iff$ [[epimorphism]], the condition of being a left resp. right inverse is stronger than being a [[monomorphism]] resp. [[epimorphism]]. This result provides a true characterization of left and right inverses in $R$-$\mathsf{Mod}$. ----- #### ---- #### 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 > ```