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
Famous quotes containing the words primary and/or ideal:
“But the doctrine of the Farm is merely this, that every man ought to stand in primary relations to the work of the world, ought to do it himself, and not to suffer the accident of his having a purse in his pocket, or his having been bred to some dishonorable and injurious craft, to sever him from those duties.”
—Ralph Waldo Emerson (1803–1882)
“In one sense it is evident that the art of kingship does include the art of lawmaking. But the political ideal is not full authority for laws but rather full authority for a man who understands the art of kingship and has kingly ability.”
—Plato (428–348 B.C.)