characterization of primary ideals

PropositionPlanetmathPlanetmath. Let R be a commutative ring and IR an ideal. Then I is primary if and only if every zero divisorMathworldPlanetmath in R/I is nilpotentPlanetmathPlanetmath.

Proof. ,,” Assume, that we have xR such that x+I is a zero divisor in R/I. In particular x+I0+I and there is yR, y+I0+I such that


This is if and only if xyI. Thus either yI or xnI for some n. Of course yI, because y+I0+I and thus xnI. Therefore xn+I=0+I, which means that x+I is nilpotent in R/I.

,,” Assume that for some x,yR we have xyI and x,yI. Then


so both x+I and y+I are zero divisors in R/I. By our assumptionPlanetmathPlanetmath both are nilpotent, and therefore there is n,m such that xn+I=ym+I=0+I. This shows, that xnI and ymI, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title characterization of primary ideals
Canonical name CharacterizationOfPrimaryIdeals
Date of creation 2013-03-22 19:04:29
Last modified on 2013-03-22 19:04:29
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Derivation
Classification msc 13C99