[Verse 1]
In the kingdom of rings where structures reign supreme
There's a ladder of power, a mathematical dream
At the bottom we find domains integral and true
No zero divisors allowed in this crew
But climb up higher and you'll discover the game
Each level adds magic to the algebraic name
[Chorus]
ED to PID to UFD, climbing the tower of certainty
Euclidean, Principal, Unique Factorization
Each one nested like Russian dolls in formation
The more like integers, the higher you go
From division algorithms to factors that flow
[Verse 2]
Euclidean domains sit upon the throne
With norm functions measuring what we own
Take the integers with absolute size
Or polynomials where degree applies
Gaussian integers with their complex dance
A squared plus B squared gives division a chance
[Chorus]
ED to PID to UFD, climbing the tower of certainty
Euclidean, Principal, Unique Factorization
Each one nested like Russian dolls in formation
The more like integers, the higher you go
From division algorithms to factors that flow
[Verse 3]
Principal domains drop the norm constraint
But every ideal has one generator saint
While unique factorization keeps the dream alive
Every element breaks down in just one way to thrive
But Z bracket X shows us the cracks appear
Two comma X can't be principal here
[Bridge]
Z root negative five breaks the sacred rule
Six equals two times three, that's the usual tool
But one plus root five times one minus the same
Gives six another way, factorization's not a game
When uniqueness crumbles, chaos takes the wheel
Algebraic number theory shows what's truly real
[Chorus]
ED to PID to UFD, climbing the tower of certainty
Euclidean, Principal, Unique Factorization
Each one nested like Russian dolls in formation
The more like integers, the higher you go
From division algorithms to factors that flow
[Outro]
Measure how Z-like your ring can be
In this hierarchy of machinery