Prime Spectra and Primary Decomposition

The space Spec R

The Nullstellensatz matched maximal ideals with points. Grothendieck's insight was to keep all primes. The prime spectrum Spec R is the set of all prime ideals of R, topologized by declaring the closed sets to be V(I) = { P prime : I ⊆ P }. This is the Zariski topology: closed sets are common-vanishing loci of collections of functions.

Non-maximal primes are the genuinely new points. A prime P that is not maximal is a generic point: its closure is the whole irreducible piece V(P), not just {P}. This is what lets a single point "spread out" over a curve or surface — the foundation of scheme theory.

Spec Z:  primes are (0) and (p) for each prime number p.
  - Each (p) is a closed point: V((p)) = {(p)}.
  - (0) is the generic point: its closure is ALL of Spec Z,
    because (0) is contained in every prime. "Z is irreducible."

Spec k[x] (k alg. closed):  primes are (0) and (x - a) for a in k.
  - Closed points (x - a)  <->  points a of the affine line A^1.
  - (0)  =  generic point of the whole line.
So Spec k[x] = the affine line PLUS one extra fuzzy point living everywhere.

Generic points: (0) sits inside every prime, so its closure fills the whole irreducible space.

Primary ideals: prime powers, done right

In Z, n = p_1^{e_1}…p_k^{e_k} translates to (n) = ∩ (p_i^{e_i}). We want this for general Noetherian rings. The right "prime power" is a primary ideal: Q is primary if ab ∈ Q forces a ∈ Q or b^n ∈ Q for some n. Equivalently, in R/Q every zero-divisor is nilpotent. The radical √Q is then a prime P, and we call Q P-primary.

The Lasker–Noether theorem

The payoff is the Lasker–Noether theorem: in a Noetherian ring, every ideal I has a primary decomposition I = Q_1 ∩ … ∩ Q_r, a finite intersection of primary ideals. After removing redundancy and grouping primaries with the same radical, the set of radicals P_i = √Q_i is uniquely determined by I. These P_i are the associated primes of I — the geometric components plus the embedded ones.

Example in k[x,y]:  I = (x^2, x*y).
  Factor the geometry: V(I) = { x = 0 } u { (0,0) } -- the y-axis,
  plus the origin appearing 'with extra thickness'.
  A primary decomposition:
      I = (x)  n  (x^2, y).
   - (x) is prime, radical (x)      -> the y-axis  (minimal prime)
   - (x^2, y) is (x,y)-primary,      -> the origin (EMBEDDED prime)
  Associated primes: { (x), (x,y) }.  (x) is minimal; (x,y) is embedded.

Note the embedded component is NOT unique: 
  I = (x) n (x^2, x*y, y^2) is another valid decomposition.
The radicals {(x),(x,y)} are unique; the primary pieces at embedded primes are not.

Minimal vs embedded primes; the set of associated primes is canonical, the primary pieces are not.

Honest warning: primary decomposition is the one classical theorem here whose uniqueness is partial. Minimal primes (the components of V(I)) are forced, but embedded primes carry non-unique primary pieces. Existence is a clean Noetherian induction; the nuance is entirely in what is and isn't canonical.