Localization: inverting on purpose
The field of fractions inverted *every* nonzero element. Localization is the controlled version: pick a multiplicative set S (closed under products, containing 1, not containing 0) and invert exactly the elements of S. The result S⁻¹R consists of fractions r/s with s ∈ S, under the relation r/s = r'/s' iff t(rs' − r's) = 0 for some t ∈ S. The extra t is the careful fix that makes this an equivalence even when R has zero divisors.
The most important choice is S = R∖P for a prime ideal P: inverting everything *outside* P. The result, written R_P, is the localization at P, and it is a local ring with unique maximal ideal P·R_P. Geometrically you have zoomed in on the point P, discarding everything that does not vanish there. Taking S = R∖{0} when R is a domain (i.e. P = (0)) recovers the field of fractions — so Frac is just localization at the zero prime.
Localizing Z at the prime (5): S = Z ∖ (5) = integers NOT divisible by 5.
Z_(5) = { a/b ∈ Q : 5 ∤ b }.
Units of Z_(5): all a/b with 5 ∤ a and 5 ∤ b.
Unique maximal ideal: 5·Z_(5) = { a/b : 5 | a, 5 ∤ b }.
Z_(5)/5Z_(5) ≅ F_5 (residue field at the point).
Ideals of Z_(5): (0) ⊂ (5) ⊂ (1)=Z_(5) — only ONE nonzero prime.
So Z_(5) has Krull dimension 1 with a single closed point: it is in fact
a discrete valuation ring (DVR), the local model of a smooth curve point.The Noetherian condition
A ring is Noetherian if it satisfies the ascending chain condition on ideals: every chain I₁ ⊆ I₂ ⊆ I₃ ⊆ … stabilizes. Three conditions are equivalent and worth having all three at hand: (i) ACC on ideals; (ii) every nonempty set of ideals has a maximal element; (iii) every ideal is finitely generated. The equivalence (i)⇔(iii) is the one you use constantly — Noetherian means "no infinitely complicated ideals".
Proof that (iii) finitely generated ⇒ (i) ACC.
Given a chain I₁ ⊆ I₂ ⊆ … let I = ⋃ Iₙ. Union of a chain of ideals
is an ideal. By (iii), I = (a₁,…,a_k) is finitely generated.
Each generator aⱼ lies in some I_{n(j)}; let N = max n(j).
Then all generators lie in I_N, so I ⊆ I_N ⊆ I, giving I = I_N.
Hence Iₙ = I_N for all n ≥ N: the chain stabilizes. ∎
Non-example: the polynomial ring in INFINITELY many variables
k[x₁, x₂, x₃, …] is NOT Noetherian:
(x₁) ⊊ (x₁,x₂) ⊊ (x₁,x₂,x₃) ⊊ … never stabilizes.Hilbert's basis theorem, and why it matters
Noetherianity propagates. Any quotient of a Noetherian ring is Noetherian (ideals of R/I lift to ideals of R), and any localization S⁻¹R of a Noetherian ring is Noetherian. The deepest closure property is the Hilbert basis theorem: if R is Noetherian then R[x] is Noetherian. By induction R[x₁,…,xₙ] is Noetherian, and then so is every finitely generated algebra over a field or over Z, since these are quotients of polynomial rings.
Why care? Almost every ring you meet in algebraic geometry and number theory — coordinate rings of varieties, rings of integers of number fields, completions — is Noetherian, and the condition is exactly what licenses the finiteness arguments those subjects depend on: ideals have finite generating sets, varieties are cut out by finitely many equations, and induction on chains terminates. Noetherian is the hypothesis that makes commutative algebra a working tool rather than a zoo of pathologies.