Order, lattices, ordered algebraic structures Definition
Ascending chain condition
Definition
Let be a partially ordered set. The set satisfies the ascending chain condition if every sequence of elements of is eventually constant. That is, there is an integer with for all .
Remarks
Equivalently, every nonempty subset of has a maximal element. This equivalence uses the axiom of dependent choice. The set with its usual order does not satisfy the condition. With the reverse order, it does. For the set of the ideals of a ring, ordered by inclusion, the condition defines Noetherian rings. The ideals of satisfy the condition: a chain means , , and so on.
Where this leads
24 pages build on this one.
- Noetherian ring
- Hilbert's basis theorem
- Regular local rings are UFDs
- Algebraic set
- Complete intersection
- Artin–Tate lemma
- Lasker–Noether theorem
- Gröbner basis
- Jacobson's conjecture
- Akizuki–Hopkins–Levitzki theorem
- Krull–Akizuki theorem
- Nagata's criterion for factoriality
- Mori–Nagata theorem
- Embedded prime
- Noetherian module
- Levitzki's theorem
- Lenagan's theorem
- Zero divisors are the union of associated primes
- Krull's principal ideal theorem
- Krull's height theorem
- Cohen's theorem on Noetherian rings
- Hopkins–Levitzki theorem
- Finiteness of associated primes
- Buchberger's algorithm
Further reading
About this page
| Tag | 0ZCAXY9563 |
|---|---|
| URI | m2://0ZCAXY9563 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 06A06 |
| Defines | Ascending chain condition |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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