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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, specifically commutative algebra, an ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then either x or yn is also an element of Q, for some n. Every prime ideal is primary. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary.