primary ideal

An ideal Q in a commutative ring R is a primary ideal if for all elements x,yR, we have that if xyQ, then either xQ or ynQ for some n.

This is clearly a generalizationPlanetmathPlanetmath of the notion of a prime idealMathworldPlanetmathPlanetmathPlanetmath, and (very) loosely mirrors the relationship in between prime numbers and prime powers.

It is clear that every prime ideal is primary.

Example. Let Q=(25) in R=. Suppose that xyQ but xQ. 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 radicalPlanetmathPlanetmath 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