prime ideal

Think of how a prime number behaves: if a prime p divides a product a·b, then it must divide a or it must divide b. A prime ideal lifts exactly this splitting behavior from numbers to ideals. It is the ring-theoretic abstraction of primality, and it is the more fundamental of the two great families of ideals (the other being maximal ideals).

Formally, in a commutative ring R with identity, a proper ideal P (so P ≠ R) is prime if whenever a·b lies in P, then a lies in P or b lies in P. An equivalent and very useful characterization: P is prime exactly when its complement R \ P is multiplicatively closed (closed under products and containing 1), and another equivalent statement is that the quotient ring R/P is an integral domain. These three views — divisibility, multiplicative complement, domain quotient — are the same fact wearing three hats.

Prime ideals are the points of algebraic geometry: the prime spectrum Spec(R) is the set of all prime ideals, and this is the foundation of scheme theory. A caveat worth stating: in a noncommutative ring the definition must be phrased with ideals rather than elements (an ideal P is prime if A·B ⊆ P forces A ⊆ P or B ⊆ P for ideals A, B), since the elementwise version is too weak. The zero ideal is prime precisely when R is an integral domain.