----- Let $R$ be a [[ring]]. > [!proposition] Proposition. ([[integral domain is a field iff its finitely generated modules are all free]]) > If $R$ is an [[integral domain]], then it is a [[field]] if and only if every [[submodule generated by a subset|finitely generated]] $R$-[[module]] is [[free module|free]]. ^proposition > [!proof]- Proof. ([[integral domain is a field iff its finitely generated modules are all free]]) > ~ > $\text{field}\implies \text{free}$ is just restating that [[every vector space has a basis|every vector space is free as a module]]. So we just need to show $\text{free} \implies \text{field}$. > Okay, suppose every finitely generated $R$-[[module]] is a [[free module]]. Then, in particular, every [[cyclic module]] over $R$ is a [[free module]]. Hence every [[cyclic module]] over $R$ is [[torsion element of a module|torsion-free]]. So we are done, by [[integral domain is a field iff its cyclic modules are torsion-free]]. ----- #### ---- #### 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 > ```