fundamental theorem of ideal theory

Theorem.โ€‰ Every nonzero ideal of the ring of integersMathworldPlanetmath of an algebraic number fieldMathworldPlanetmath can be written as product ( of prime idealsMathworldPlanetmathPlanetmathPlanetmath of the ring.โ€‰ The prime ideal of the factors (

In this entry we consider the ring ๐’ช of the integers of a number field โ„šโข(ฯ‘).โ€‰ We use as starting the fact that the ideals of ๐’ช are finitely generatedMathworldPlanetmathPlanetmath submodulesMathworldPlanetmath of ๐’ช (cf. basis of ideal in algebraic number field) and that its prime ideals ๐”ญ are maximal idealsMathworldPlanetmath, i.e. the only ideal factors of ๐”ญ are ๐”ญ itself and the unit ideal โ€‰(1)=๐’ช.

For proving the above fundamental theorem of ideal theory, we present and prove some lemmata.

Lemma 1.โ€‰ The equation โ€‰๐”ž=๐”Ÿโข๐” โ€‰ between the ideals of ๐’ช implies thatโ€‰ ๐”žโŠ†๐” .

Proof.โ€‰ Letโ€‰ ๐”Ÿ=(ฮฒ1,โ€ฆ,ฮฒs)โ€‰ andโ€‰ ๐” =(ฮณ1,โ€ฆ,ฮณt).โ€‰ If


then there are the elements ฮปiโขj of ๐’ช such that


But the of ฮณj in the parentheses are elements of the ring ๐’ช, whence the last sum form of ฮฑ shows thatโ€‰ ฮฑโˆˆ๐” .โ€‰ Consequently, ๐”žโŠ†๐” .

Lemma 2.โ€‰ Any nonzero element ฮฑ of ๐’ช belongs only to a finite number of ideals of ๐’ช.

Proof.โ€‰ Letโ€‰ ๐”ž=(ฮฑ1,โ€ฆ,ฮฑr)โ€‰ be any ideal containing ฮฑ and letโ€‰ {ฯฑ1,โ€ฆ,ฯฑm}โ€‰ be a complete residue systemMathworldPlanetmath modulo ฮฑ (cf. congruence in algebraic number field).โ€‰ Then


where the numbers ฮปi belong to ๐’ช.โ€‰ Since we have


there can be different ideals ๐”ž only a finite number, at most 1+m+(n2)+โ€ฆ+(mm)=2m.

Lemma 3.โ€‰ Each ideal ๐”ž of ๐’ช has only a finite number of ideal factors.

Proof.โ€‰ If โ€‰๐” โˆฃ๐”žโ€‰ andโ€‰ ฮฑโˆˆ๐”ž,โ€‰ then by Lemma 1,โ€‰ ฮฑโˆˆ๐” ,โ€‰ whence Lemma 2 implies that there is only a finite number of such factors ๐” .

Lemma 4.โ€‰ All nonzero ideals of ๐’ช are cancellativePlanetmathPlanetmath (, i.e. ifโ€‰ ๐”žโข๐” =๐”žโข๐”กโ€‰ thenโ€‰ ๐” =๐”ก.

Proof.โ€‰ The theorem of (1911) guarantees an ideal ๐”ค of ๐’ช such that the product ๐”คโข๐”ž is a principal idealMathworldPlanetmathPlanetmathPlanetmath (ฯ‰).โ€‰ Then we may write

(ฯ‰)โข๐” =(๐”คโข๐”ž)โข๐” =๐”คโข(๐”žโข๐” )=๐”คโข(๐”žโข๐”ก)=(๐”คโข๐”ž)โข๐”ก=(ฯ‰)โข๐”ก.

Ifโ€‰ ๐” =(ฮณ1,โ€ฆ,ฮณs)โ€‰ andโ€‰ ๐”ก=(ฮด1,โ€ฆ,ฮดt),โ€‰ we thus have the equation


by which there must exist the elements ฮปiโข1,โ€ฆ,ฮปiโขt of ๐’ช such that


Consequently, the โ€‰ฮณi=ฮปiโข1โขฮด1+โ€ฆ+ฮปiโขtโขฮดtโ€‰ of ๐”  belong to the ideal ๐”ก, and therefore โ€‰๐” โŠ†๐”ก.โ€‰ Similarly one gets the reverse containment.

Lemma 5.โ€‰ Ifโ€‰ ๐”ž=๐”Ÿโข๐” โ€‰ andโ€‰ ๐”Ÿโ‰ (1),โ€‰ then ๐”  has less ideal factors than ๐”ž.

Proof.โ€‰ Evidently, any factor of ๐”  is a factor of ๐”ž.โ€‰ Butโ€‰ ๐”žโˆฃ๐”žโ€‰ andโ€‰ ๐”žโˆค๐” , since otherwise we hadโ€‰ ๐” =๐”žโข๐”ก=๐”Ÿโข๐” โข๐”กโ€‰ whence (1)=๐”Ÿโข๐”ก which would, by Lemma 4, imply ๐”Ÿ=(1).

Lemma 6.โ€‰ Any proper idealMathworldPlanetmath ๐”ž of ๐’ช has a prime ideal factor.

Proof.โ€‰ Let ๐”  be such a factor of ๐”ž that has as few factors as possible.โ€‰ Then ๐”  must be a prime ideal, because otherwise we hadโ€‰ ๐” =๐” 1โข๐”กโ€‰ where ๐” 1 and ๐”ก are proper ideals of ๐’ช and, by Lemma 5, the ideal ๐” 1 would have less factors than ๐” ; this however contradicts the factโ€‰ ๐” 1โˆฃ๐”ž.

Lemma 7.โ€‰ Every nonzero proper ideal ๐”ž of ๐’ช can be written as a product ๐”ญ1โขโ‹ฏโข๐”ญk whereโ€‰ k>0โ€‰ and the factors ๐”ญi are prime ideals.

Proof.โ€‰ If ๐”ž has only one factor ๐”ญ distinct from (1), thenโ€‰ ๐”ž=๐”ญโ€‰ is a prime ideal.
Induction hypothesis:โ€‰ Lemma 7 is in always when ๐”ž has at most n factors.โ€‰ Let ๐”ž now have n+1 factors.โ€‰ Lemma 6 implies that there is a prime ideal ๐”ญ such thatโ€‰ ๐”ž=๐”ญโข๐”กโ€‰ whereโ€‰ ๐”กโ‰ (1)โ€‰ and ๐”ก has, by Lemma 5, at most n factors.โ€‰ Hence,โ€‰ ๐”ก=๐”ญ1โขโ‹ฏโข๐”ญkโ€‰ and therefore,โ€‰ ๐”ž=๐”ญโข๐”ญ1โขโ‹ฏโข๐”ญkโ€‰ where all ๐”ญโ€™s are prime ideals.

Lemma 8.โ€‰ Any two prime factorMathworldPlanetmathPlanetmath


of a nonzero ideal ๐”ž of ๐’ช are identical, i.e.โ€‰ r=sโ€‰ and each prime factor ๐”ญi is equal to a prime factor ๐”ฎj and vice versa.

Proof.โ€‰ Any prime ideal has the property that if it divides a product of ideals, it divides one of the factors of the product; now these factors are prime ideals and therefore the prime ideal coincides with one of the factors.โ€‰ Similarly as in the proof of the fundamental theorem of arithmetics, one sees the uniqueness of the prime factorisation of ๐”ž.

Title fundamental theorem of ideal theory
Canonical name FundamentalTheoremOfIdealTheory
Date of creation 2013-03-22 19:12:40
Last modified on 2013-03-22 19:12:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 27
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Synonym principal theorem of ideal theory
Related topic AlgebraicNumberTheory
Related topic DedekindDomain
Related topic EveryIdealInADedekindDomainIsAFactorOfAPrincipalIdeal
Related topic PrimeIdealFactorizationIsUnique
Related topic UniqueFactorizationAndIdealsInRingOfIntegers
Related topic CancellativeSemigroup