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":
... 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 ...
Famous quotes containing the words ideal and/or primary:
“The tradition I cherish is the ideal this country was built upon, the concept of religious pluralism, of a plethora of opinions, of tolerance and not the jihad. Religious war, pooh. The war is between those who trust us to think and those who believe we must merely be led.”
—Anna Quindlen (b. 1952)
“One of the effects of a safe and civilised life is an immense oversensitiveness which makes all the primary emotions somewhat disgusting. Generosity is as painful as meanness, gratitude as hateful as ingratitude.”
—George Orwell (19031950)