[Verse 1]
Start with a module M, elements dancing free
Take a subset N that holds the same degree
Closed under addition, scalar multiplication too
Now you've got a submodule, split in two
[Chorus]
Submodules nested, quotient classes born
M over N, equivalence worn
First theorem singing ker phi to im
Second shows the dance of A plus B within
Third cascades quotients stacked up high
Module isomorphisms never lie
[Verse 2]
Quotient module M slash N appears
Cosets m plus N, shifting gears
Operations defined on representatives
Addition stays consistent, multiplication gives
[Chorus]
Submodules nested, quotient classes born
M over N, equivalence worn
First theorem singing ker phi to im
Second shows the dance of A plus B within
Third cascades quotients stacked up high
Module isomorphisms never lie
[Bridge]
A plus B over B isomorphic clean
To A over A intersect B's scene
Stack quotients up like Russian dolls
M over A, then B over A calls
[Verse 3]
Homomorphism phi maps M away
Kernel kills to zero, image holds the sway
M over ker phi equals im phi true
Same patterns flow from groups we knew
[Chorus]
Submodules nested, quotient classes born
M over N, equivalence worn
First theorem singing ker phi to im
Second shows the dance of A plus B within
Third cascades quotients stacked up high
Module isomorphisms never lie
[Outro]
Groups to rings to modules, theorems stay the same
Algebraic structures playing the same game
Quotients reveal the hidden symmetry
In every mathematical poetry