Prime ideal

A proper ideal \(P\) of a commutative ring such that \(ab\in P\) implies \(a\in P\) or \(b\in P\). Its quotient ring has no zero divisors.

Ring: A set equipped with addition and multiplication satisfying the ring axioms. Prime ideals are defined inside rings, whose multiplication determines the condition.
Prime element: A nonzero nonunit element whose divisibility of a product forces it to divide a factor. In a principal ideal domain, an element is prime precisely when its generated ideal is prime.
Spectrum of a ring: The set of prime ideals of a commutative ring, equipped with the Zariski topology. Each prime ideal becomes a point in the ring's spectrum.
Primary ideal: A proper ideal whose product containment forces one factor to have a power in the ideal. Primary ideals weaken primality by allowing powers of a factor rather than the factor itself.
Ideal: An additive subgroup of a ring closed under multiplication by every ring element. Primality is an additional condition imposed on an ideal.
Radical of an ideal: The ideal of elements some positive power of which belongs to a given ideal. Every prime ideal is radical, linking product detection to power detection.
Zariski topology: A topology whose closed sets in algebraic geometry are defined by polynomial equations. Prime ideals determine the basic closed sets in the spectrum of a ring.
Irreducible ideal: A proper ideal that cannot be expressed as the intersection of two strictly larger ideals. Irreducibility concerns intersections, unlike primality's condition on products.
Quotient ring: A ring formed by identifying elements whose difference belongs to a chosen ideal. Factoring out a prime ideal produces a ring without zero divisors.
Localization of a ring: A ring constructed by making a chosen multiplicative set of elements invertible. A prime ideal disjoint from the inverted set extends to a prime ideal after localization.

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