---- > [!definition] Definition. ([[rank of a finitely generated module]]) > Let $R$ be an [[integral domain]]. The **rank** $\text{rk }M$ of a [[submodule generated by a subset|finitely generated]] $R$-[[module]] $M=\langle m_{1},\dots,m_{n} \rangle$ is the natural number $\text{max}_{A} |A|$ > where $A$ ranges over all [[linearly independent]] subsets of $M$. ^definition > [!justification] > Show this number indeed is always finite. ^justification ---- #### ---- #### 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 > ```