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单词 CharacterizationOfPrimaryIdeals
释义

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

0+I=(x+I)(y+I)=xy+I.

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

(x+I)(y+I)=xy+I=0+I,

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.

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更新时间:2025/5/4 14:36:02