Ring theory MOC

Noetherian ring

A ring 𝑅 is called (left) Noetherian iff any of the following equivalent conditions hold:¹ ring

1. every (left) ideal 𝐼 ⊴𝑅 is finitely generated as a (left) 𝑅-module, i.e. 𝑅 is a (left) Noetherian module;
2. (ascending chain condition or ACC) every increasing sequence 𝐼1 ⊴𝐼2 ⊴⋯ of (left) ideals of 𝑅 has a largest element;
3. every non-empty set of (left) ideals of 𝑅 contains a maximal element.

Proof

Suppose ^N1 holds, and let 𝐼1 ⊴𝐼2 ⊴⋯ be an increasing sequence of (left/right) ideals. Then

𝐼=∞⋃𝑖=1𝐼𝑛

is an ideal, since if 𝑥 ∈𝐼 then 𝑥 ∈𝐼𝑗 for some 𝑗 and thus 𝑟𝑥 ∈𝐼𝑗 resp. 𝑥𝑟 ∈𝐼𝑗 for any 𝑟 ∈𝑅. Hence 𝐼 is finitely generated, and all these generators must be exhausted by some 𝐼𝑛, implying ^N2.

Suppose ^N2 holds, and assume towards contradiction there exists a set ℑ of (left/right) ideals with no maximal element. One can then form a strictly increasing sequence of 𝐼𝑛 ∈ℑ, contradicting ^N2. Therefore ^N2 implies ^N3.

Suppose ^N3 holds, and let 𝐼 ⊴𝑅 be an arbitrary (left/right) ideal. Letting ℑ be the set of finitely generated ideals contained in 𝐼, which is inhabited since 0 ∈ℑ, and thus contains a maximal element 𝐼′ ∈ℑ. Assume towards contradiction 𝐼′ ≠𝐼. Then we can take 𝑥 ∈𝐼 ∖𝐼′, and form 𝐼 +𝑅𝑥 resp. 𝐼 +𝑥𝑅, which is a finitely generated (left/right) ideal strictly larger than 𝐼′, contradicting maximality. Therefore 𝐼′ =𝐼 and 𝐼 is finitely generated, so ^N3 implies ^N1.

Properties

Let 𝑅 be two-sided Noetherian.

1. Let 𝐼 ◃𝑅 be a nonzero proper ideal. Then there exist nonzero prime ideals 𝔭1,…,𝔭𝑛 ◃𝑅 such that 𝔭1⋯𝔭𝑛 ⊆𝐼.

Proof

Let I be the set of all ideals for which ^P1 fails, and assume towards contradiction its maximal element is 𝐼 ∈I, which cannot be a prime ideal, so there exist 𝑎,𝑏 ∈𝑅 ∖𝐼 such that 𝑎𝑏 ∈𝐼. Let 𝔞 =(𝐼,𝑎) and 𝔟 =(𝐼,𝑏), whence 𝐼 ⊊𝔞,𝔟, so by maximality 𝔞,𝔟 ∉I and thus there exist nonzero prime ideals 𝔭1,…,𝔭𝑚,𝔮1,…𝔮𝑛 ◃𝑅 such that

𝔞⊆(𝔭1⋯𝔭𝑚),𝔟⊆(𝔮1⋯𝔮𝑛).

Then

(𝔞𝔟)=(𝐼2,𝑎𝐼,𝑏𝐼,𝑎𝑏)⊆𝐼

whence

(𝔭1⋯𝔭𝑚𝔮1⋯𝔮𝑛)⊆𝐼,

a contradiction. Therefore I =∅.

Other results

1. Finitely generated modules over a noetherian ring are noetherian (^P2)

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Footnotes

1. 2022. Algebraic number theory course notes, §2.5, pp. 14–15 ↩