Jump to content

Ascending chain condition

From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by 85.130.107.83 (talk) at 19:37, 7 October 2012 (Chain actually has the more general meaning of totally ordered set). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

The ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings.[1][2][3] These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

Definition

A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if every strictly ascending sequence of elements eventually terminates. Equivalently, given any sequence

there exists a positive integer n such that

Similarly, P is said to satisfy the descending chain condition (DCC) if every strictly descending sequence of elements eventually terminates, or equivalently if any descending sequence

of elements of P eventually stabilizes (that is, there is no infinite descending chain).

Comments

  • A subtly different and stronger condition than "containing no infinite ascending/descending chains" is "contains no arbitrarily long ascending/descending chains (optionally, 'based at a given element')". For instance, the disjoint union of the posets {0}, {0,1}, {0,1,2}, etc., satisfies both the ACC and the DCC, but has arbitrarily long chains. If one further identifies the 0 in all of these sets, then every chain is finite, but there are arbitrarily long chains based at 0.
  • The descending chain condition on P is equivalent to P being well-founded: every nonempty subset of P has a minimal element (also called the minimal condition).
  • Similarly, the ascending chain condition is equivalent to P being converse well-founded: every nonempty subset of P has a maximal element (the maximal condition).
  • Every finite poset satisfies both ACC and DCC.

See also

Notes

  1. ^ Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.
  2. ^ Fraleigh & Katz (1967), p. 366, Lemma 7.1
  3. ^ Jacobson (2009), p. 142 and 147

References

  • Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9
  • Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko. Algebras, rings and modules. Kluwer Academic Publishers, 2004. ISBN 1-4020-2690-0
  • John B. Fraleigh, Victor J. Katz. A first course in abstract algebra. Addison-Wesley Publishing Company. 5 ed., 1967. ISBN 0-201-53467-3
  • Nathan Jacobson. Basic Algebra I. Dover, 2009. ISBN 978-0-486-47189-1