primary ideal
An ideal Q in a commutative ring R is a primary ideal if for all elements x,y∈R, we have that if xy∈Q, then either x∈Q or yn∈Q for some n∈ℕ.
This is clearly a generalization of the notion of a prime ideal
, and (very) loosely mirrors the relationship in ℤ between prime numbers and prime powers.
It is clear that every prime ideal is primary.
Example. Let Q=(25) in R=ℤ. Suppose that xy∈Q but x∉Q. 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.
The radical of a primary ideal is always a prime ideal. If P is the radical of the primary ideal Q, we say that Q is P-primary.
Title | primary ideal |
---|---|
Canonical name | PrimaryIdeal |
Date of creation | 2013-03-22 14:15:01 |
Last modified on | 2013-03-22 14:15:01 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 6 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 13C99 |
Defines | primary |
Defines | P-primary |