Math 2.0

Order, lattices, ordered algebraic structures Definition

Descending chain condition

Builds on Partial orderSequenceWell-founded orderAscending chain conditionArtinian ring

Definition

Let (P,≤)(P, \le) be a partially ordered set. The poset PP satisfies the descending chain condition when every sequence x1≥x2≥x3≥⋯x_1 \ge x_2 \ge x_3 \ge \cdots in PP is eventually constant. That is, there is an index nn such that xm=xnx_m = x_n for all m≥nm \ge n.

Remarks

Example: the ideals of the ring ℤ/12ℤ\Z / 12\Z, ordered by inclusion, satisfy the descending chain condition. Check: there are only six ideals, so a decreasing sequence of ideals takes finitely many values and must be eventually constant. Rings whose ideals satisfy the condition are the Artinian rings.

Non-example: the ideals of ℤ\Z do not satisfy it, since 2ℤ⊋4ℤ⊋8ℤ⊋⋯2\Z \supsetneq 4\Z \supsetneq 8\Z \supsetneq \cdots never stops.

The condition is the ascending chain condition for the opposite order. A poset satisfies it if and only if every nonempty subset has a minimal element, that is, the order is well-founded. The direction from the descending chain condition to minimal elements uses the axiom of dependent choice. The abbreviation DCC is common, and the minimal-element form is called the minimal condition.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Artinian ring, concept, no evidence levelArtinian ringconceptAscending chain condition, concept, no evidence levelAscending chain condi…conceptPartial order, concept, no evidence levelPartial orderconceptSequence, concept, no evidence levelSequenceconceptWell-founded order, concept, no evidence levelWell-founded orderconceptDescending chain condition, definition, no evidence levelDescending chain cond…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Artinian ring concept
  2. Ascending chain condition concept
  3. Partial order concept
  4. Sequence concept
  5. Well-founded order concept
  6. Descending chain condition definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Ascending_chain_condition
About this page
TagXRGSKJVF0M
URIm2://XRGSKJVF0M
Kinddefinition
WritingW0 stub
MSC06A06, 13E10
DefinesDescending chain condition
UsesPartial order, Sequence, Well-founded order, Ascending chain condition, Artinian ring
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.