Primary Ideal

Primary Ideal

In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n>0. 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.

Various methods of generalizing primary ideals to noncommutative rings exist but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.

Read more about Primary Ideal:  Examples and Properties

Other articles related to "primary ideal, ideal, primary":

Primary Ideal - Examples and Properties
... Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime ... Every primary ideal is primal ... If Q is a primary ideal, then the radical of Q is necessarily a prime ideal P, and this ideal is called the associated prime ideal of Q ...

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