Primary ideal
From Wikipedia, the free encyclopedia
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 
Alternatively, an ideal 
is primary if and only if 
and every zero divisor is nilpotent.
This is a generalization of the notion of a prime ideal (thus every prime ideal is primary), and (very) loosely mirrors the relationship in 
between prime numbers and prime powers.
If the radical of the primary ideal Q is the prime ideal P, then Q is said to be P-primary.
[edit] Example
Let Q = (125) in 
Suppose that but 
Then 125 | xy, but 125 does not divide x. Thus 5 must divide y, so some power of y (namely, y3), must be in Q. Therefore Q is primary.
[edit] See also
This article incorporates material from Primary ideal on PlanetMath, which is licensed under the GFDL.
