Primidealen

Primidealen, or prime ideals, are a fundamental concept in ring theory and algebraic geometry. Let R be a commutative ring with unity. A proper ideal P ⊆ R is prime if whenever ab ∈ P, then either a ∈ P or b ∈ P. Equivalently, the quotient ring R/P is an integral domain.

Properties. Every maximal ideal is prime, but not every prime ideal is maximal in general. The radical of an ideal I equals the intersection of all prime ideals containing I; in particular, the nilradical (the set of nilpotent elements) equals the intersection of all prime ideals of R. In Noetherian rings, the structure of prime ideals leads to Krull dimension, defined as the supremum of the lengths of chains of prime ideals.

Examples. In the ring of integers Z, the prime ideals are (0) and (p) for primes p. In a polynomial ring over a field, F[x], prime ideals are (0) and (f) where f is irreducible. In two variables over an algebraically closed field k, maximal ideals often have the form (x − a, y − b); more generally, maximal ideals correspond to algebraic points under suitable conditions, while prime ideals correspond to irreducible algebraic sets.

Spec(R). The set of all prime ideals of R is called Spec(R), equipped with the Zariski topology. Closed sets are defined by V(I) = {P ∈ Spec(R) : I ⊆ P}. Spec(R) provides a bridge between algebra and geometry: irreducible closed subsets correspond to prime ideals, and geometric intuition often guides algebraic reasoning.

Applications. Prime ideals are central to algebraic geometry, scheme theory, and number theory, serving as the algebraic underpinnings for studying solutions to polynomial equations and their geometric and arithmetic structure.