In mathematics, an ideal Q in a commutative ring R is a primary ideal if for all elements , we have that if , then either or for some Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, the term ideal has multiple meanings. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

This is clearly a generalization of the notion of a prime ideal, and (very) loosely mirrors the relationship in between prime numbers and prime powers. In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, a prime power is a positive integer power of a prime number. ...

Every prime ideal is primary.

Example. Let Q = (25) in Suppose that but Then 25 | xy, but 25 does not divide x. Thus 5 must divide y, and thus some power of y (namely, y²), must be in Q. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

If the radical of the primary ideal Q is the prime ideal P, then Q is said to be P-primary. In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. ...

See also: primary decomposition In mathematics, the Lasker-Noether theorem provides a vast generalization of the fundamental theorem of arithmetic to embrace the rings of algebraic geometry. ...

This article incorporates material from Primary ideal on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...