primary ideal
An ideal in a commutative ring is a primary ideal if for all elements , we have that if , then either or for some .
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.
It is clear that every prime ideal is primary.
Example. Let in . Suppose that but . Then , but 25 does not divide . Thus 5 must divide , and thus some power of (namely, ), must be in .
The radical of a primary ideal is always a prime ideal. If is the radical of the primary ideal , we say that is -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 | -primary |