----- > [!proposition] Proposition. ([[characterization of projective modules]]) > Let $R$ be a ([[commutative ring|commutative]]) [[ring]] and $P$ an $R$-[[module]]. Then the following are equivalent: > > 1. $P$ is a [[projective module|projective]] $R$-[[module]], that is, the [[hom functor|hom]] [[covariant functor|functor]] $\text{Hom}_{R\text{-}\mathsf{Mod}}(M, -)$ is [[exact functor|exact]]; > 2. For all [[epimorphism|epimorphisms]] of $R$-[[module|modules]] $\mu:M \to N$, every $R$-[[linear map|linear map]] $p:P \to N$ [[lifting|lifts]] to an $R$-[[linear map|linear map]] $\hat{p}: P \to M$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQBZEAX1PU1z5CKchWp0mrdgDkefEBmx4CRAExiaDFm0Qhic-kqFFRxcVqm6ACj3EwoAc3hFQAMwBOEALZIyIHBBIohLa7AA6YV7MIDSM9ABGMIxWAsrCIO5YDgAWOAYgHt5BNAFIqrxunj6IAMwlgYjBFjoK+YXVdf4NfoxYYC1QEDg49jEhliARMAAeWHA4cAAEEdn0OMBo3GNxicmpxrqZOXnclNxAA > \begin{tikzcd} > & P \arrow[d, "p"] \arrow[ld, "\exists \hat{p}"', dotted] & \\ > M \arrow[r, "\mu"'] & N \arrow[r] & 0 > \end{tikzcd} > \end{document} > ``` > 1. For every $R$-[[module]] $N$ and [[submodule]] $N' \subset N$, every $R$-[[linear map|linear map]] $P \to \frac{N}{N'}$ [[lifting|lifts]] to $N$, i.e., factors as $P \xrightarrow{g} N \xrightarrow{\pi} \frac{N}{N'}$ for some $R$-linear $g$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkQBfU9TXfIRQBGUsKq1GLNgDkA9DIDk3XiAzY8BIqMrV6zVohAzuEmFADm8IqABmAJwgBbJGRA4ISAEx6phkBYqdo4uiD7unoiikgZsADpxaFhBIA7OrtQeSMJcFFxAA > \begin{tikzcd} > P \arrow[r, "g"] \arrow[rd] & N \arrow[d, "\pi"] \\ > & N/N' > \end{tikzcd} > \end{document} > ``` > > 2. Every [[short exact sequence]] of the form $0 \to A \to B \to P \to 0$ [[split short exact sequence|splits]]; > 3. There is an $R$-[[module]] $K$ such that $P \oplus K$ is a [[free module|free]] $R$-[[module]] (i.e., $P$ is a [[direct sum of modules|direct summand]] of a [[free module]]). ----- #### [^1]: Compare to the case of [[flat module|flat modules]], where ---- #### 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 > ```