primarity

Primarity is a term that occasionally appears in abstract algebra to denote the property of an ideal or element having a primary-structure flavor. In standard mathematical usage, the corresponding and widely accepted term is primary; primarity is not widely standardized and is encountered mainly as a historical or informal synonym in some papers.

In ring theory, an ideal I of a commutative ring R with unity is primary if whenever ab ∈ I and a ∉ √I, then b^n ∈ I for some positive integer n. Equivalently, the quotient ring R/I has a unique associated prime, and the radical √I is a prime ideal. Primary ideals play a central role in primary decomposition, where (under suitable conditions) an ideal is expressed as an intersection of primary ideals.

Context and examples. In the ring of integers Z, every ideal of the form (p^k) with p prime and k ≥ 1 is primary, with radical √(p^k) = (p). More generally, in polynomial and other Noetherian rings, primary ideals capture powers of prime-like behavior around a given prime ideal.

Relationship to primeness and decomposition. Primarity is stronger than being merely contained in a prime ideal and weaker than being prime itself. A primary decomposition expresses an ideal as an intersection of primary ideals, each associated to a prime ideal. The radicals of these primary components are the associated primes of the original ideal.

Relation to terminology. While primarity conveys the idea of primary-ness, the formal and widely accepted terminology remains primary, primary ideal, and primary decomposition. Primarity is sometimes encountered in historical contexts or informal discourse but is not a standard technical term in modern algebra. See also: prime ideal, primary ideal, radical, primary decomposition, associated primes.