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单词 EveryPIDIsAUFDAlternativeProof
释义

every PID is a UFD - alternative proof


Proposition. If R is a principal ideal domainMathworldPlanetmath, then R is a unique factorization domainMathworldPlanetmath.

Proof. Recall, that due to Kaplansky Theorem (see this article (http://planetmath.org/EquivalentDefinitionsForUFD) for details) it is enough to show that every nonzero prime idealMathworldPlanetmathPlanetmathPlanetmath in R contains a prime elementMathworldPlanetmath.

On the other hand, recall that an element pR is prime if and only if an ideal (p) generated by p is nonzero and prime.

Thus, if P is a nonzero prime ideal in R, then (since R is a PID) there exists pR such that P=(p). This completes the proof.

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更新时间:2025/5/4 6:45:23