---- > [!theorem] Theorem. ([[the rank theorem for free modules]]) > Let $R$ be an [[integral domain]], and let $A,B$ be sets. Then $F^{R}(A) \cong F^{R}(B) \iff |A|=|B|,$ > that is, the [[free module|free]] $R$-[[module|modules]] on $A$ and $B$ are [[module isomorphism|isomorphic]] if and only if $A$ and $B$ are in [[bijection|bijective]] correspondence. ^theorem > [!NOTE] Remark. > In some sense, this can be viewed as classification of [[free module|free modules]] over (say) [[integral domain|a given integral domain]]: if $F$ is a [[free module|free]] $R$-[[module]], then there is a unique (up to [[bijection]]) set $A$ such that $F \cong R^{\oplus A}$. > > But the choice of isomorphism matters a lot! Indeed, the choice of [[coordinate isomorphism|coordinate]] [[module isomorphism|isomorphism]] amounts to a [[module is free iff admits basis|choice of]] [[basis]] of $F$. > [!specialization] > If $R=k$ is a [[field]] and $M=V$ is thus a $k$-[[vector space]], this result is known as **the dimension theorem**: two [[vector space|vector spaces]] are [[linear isomorphism|isomorphic]] if and only if their [[dimension|dimensions]] match. > > For finite-dimensional vector spaces the proof requires less machinery; this result is [[the (finite) dimension theorem]]. ^specialization ---- #### ----- #### 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 > ```