---- > [!definition] Definition. ([[projective module]]) > Let $R$ be a [[commutative ring|(say, commutative)]] [[ring]]. An $R$-[[module]] $P$ is **projective** if the [[hom functor|hom]] [[covariant functor|functor]] $\text{Hom}_{R\text{-}\mathsf{Mod}}(P, -)$ is [[exact functor|exact]].[^1] ^definition ---- #### [^1]: Compare to the notion of a [[flat module]] $N$, wherein it is required that the [[tensor functor]] $\_ \times_{R} N$ is [[exact functor|exact]]. ---- #### 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 > ```