[Verse 1]
When you've got a PID domain, R's the name
And M's your module, finitely made
There's a theorem that breaks it down complete
Into pieces that are neat and sweet
[Chorus]
Split it up, R to the r plus quotients lined
D one divides d two divides, the chain's designed
Invariant factors tell the tale
Structure theorem never fails
Every module finds its form
In this algebraic storm
[Verse 2]
Start with integers, that's your PID
Abelian groups come naturally
Z mod n components show their face
Cyclic factors find their place
[Chorus]
Split it up, R to the r plus quotients lined
D one divides d two divides, the chain's designed
Invariant factors tell the tale
Structure theorem never fails
Every module finds its form
In this algebraic storm
[Verse 3]
Switch to polynomials F of x
Jordan canonical reflects
Same theorem, different disguise
Matrix forms before your eyes
[Bridge]
One truth beneath two worlds apart
Abelian groups and linear art
The PID structure holds the key
Unifying what we see
[Chorus]
Split it up, R to the r plus quotients lined
D one divides d two divides, the chain's designed
Invariant factors tell the tale
Structure theorem never fails
Every module finds its form
In this algebraic storm
[Outro]
R direct sum quotient rings
Beauty that this theorem brings
Classification crystallized
In structure theorems' eyes