---- > [!definition] Definition. ([[rank of a free module]]) > Let $R$ be a [[integral domain]]. The **rank** of a [[free module|free]] $R$-[[module]] $M$, denoted $\text{rk}_{R} \ M$, is the [[cardinality]] of a ([[cardinality of linearly independent subset cannot exceed cardinality of maximally linearly independent subset|hence any]]) [[maximal]] [[linearly independent]] subset of $M$. > > ^definition > [!justification] > This notion is [[well-defined]] by [[cardinality of linearly independent subset cannot exceed cardinality of maximally linearly independent subset|any two maximal linearly independent subsets of a free module over an integral domain have the same cardinality]]. ^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 > ```