Prime and Maximal Ideals

Two slogans

Here are the definitions, stated as the slogans you should carry forever. A proper ideal P is prime if ab ∈ P forces a ∈ P or b ∈ P — equivalently, R/P is an integral domain. A proper ideal M is maximal if no ideal sits strictly between M and R — equivalently, R/M is a field. The translation between the ring-theoretic condition and the quotient condition is the whole content, and both directions are short.

Proof that P prime ⇔ R/P is a domain.

  ab ∈ P  ⇔  (a+P)(b+P) = ab + P = 0  in R/P.
  So "ab ∈ P ⇒ a ∈ P or b ∈ P"
     ⇔  "(a+P)(b+P)=0 ⇒ a+P=0 or b+P=0"
     ⇔  R/P has no nonzero zero divisors
     ⇔  R/P is an integral domain.            ∎

Proof that M maximal ⇔ R/M is a field.
  Ideals of R/M  ↔  ideals of R containing M   (correspondence theorem).
  M maximal ⇔ only such ideals are M and R
           ⇔ only ideals of R/M are (0) and R/M
           ⇔ R/M is a field (a comm. ring with exactly two ideals). ∎

Maximal implies prime, but not conversely

Every field is a domain, so the quotient characterizations instantly give: every maximal ideal is prime. The converse fails, and the failure is informative. In Z the prime ideals are (0) and (p) for primes p; the maximal ideals are exactly the (p). The odd one out is (0): it is prime (Z is a domain) but not maximal (it sits inside every (p)). So (0) being prime-but-not-maximal records the fact that Z is an infinite domain that is not a field.

A two-dimensional example makes the gap geometric. In R = k[x,y], the ideal (x) is prime — R/(x) ≅ k[y] is a domain — but not maximal, because (x) ⊊ (x,y) ⊊ R. Here (x,y) is maximal, with quotient k. The chain (0) ⊊ (x) ⊊ (x,y) of primes has length 2, and that length is the Krull dimension of the polynomial ring in two variables. Prime ideals that are not maximal are how a ring records that it has dimension bigger than zero.

The spectrum, and local rings

Collect all prime ideals of R into a set Spec R, the prime spectrum. This is the first object of algebraic geometry: points of Spec R are primes, the maximal ideals are the "classical" points, and the non-maximal primes are "generic" points fattening up subvarieties. A ring with exactly one maximal ideal is called a local ring; localizing a ring at a prime (a later guide) is the standard way to manufacture one, zooming in on a single point.

1. To test primality, quotient and ask "is this a domain?" — usually faster than checking ab∈P directly.
2. To test maximality, quotient and ask "is this a field?" — e.g. (x²+1) in R[x] is maximal because the quotient is C.
3. In a PID (next guide), the nonzero primes and the maximals coincide, so dimension drops to 1 and Spec is especially simple.