[Verse 1] Take an abelian group and dress it up with more Scalar multiplication knocking at the door Ring times module element, axioms align Distributive laws dancing in mathematical time Vector spaces wore the crown when fields were all we knew But rings break all the rules and modules see us through [Chorus] Modules stretch beyond the field When rings refuse to yield Every integer makes groups sing Every ideal's a ring-shaped thing Not every basis can be found Not every complement comes around Modules are the wilder breed Harder structures that we need [Verse 2] Integers multiply any abelian crew Add the element to itself, that's how the magic grew Polynomial rings with x can build a linear map Vector spaces with transformations caught up in the trap Ideals become modules when the ring acts on itself Abstract algebra's treasure sitting on the shelf [Chorus] Modules stretch beyond the field When rings refuse to yield Every integer makes groups sing Every ideal's a ring-shaped thing Not every basis can be found Not every complement comes around Modules are the wilder breed Harder structures that we need [Bridge] Free modules carry basis proud Semisimple splits allowed But most modules play by different laws No complement without a cause Structure theory runs so deep Harder patterns that we keep [Verse 3] Four axioms guard the gate like vector space before Left distributive, right distributive, two more at the core Scalar sum distributes clean, sum of scalars too Identity and associative see the journey through From simple fields to complex rings, the ladder climbs so high Modules are the reason why mathematics learns to fly [Final Chorus] Modules stretch beyond the field When rings refuse to yield Richer structure, wilder game Vector spaces can't contain All the beauty that unfolds When the ring story gets told [Outro] Ring acts left, the module waits Abstract beauty at the gates
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