Ascending chain condition: Difference between revisions

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This implies ACC+DCC but the conjunction does not imply this - hence this is stronger than ACC+DCC
 
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Undid revision 572353659 by Daira Hopwood (talk) I was mistaken
 
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In mathematics, the '''ascending chain condition (ACC)''' and '''descending chain condition (DCC)''' are finiteness properties satisfied by some algebraic structures, most importantly, [[Ideal (ring theory)|ideal]]s in certain [[commutative ring]]s.<ref>Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.</ref><ref>Fraleigh & Katz (1967), p. 366, Lemma 7.1</ref><ref>Jacobson (2009), p. 142 and 147</ref> 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 (mathematics)|eventually]] terminates. Equivalently, given any sequence
:<math>a_1 \,\leq\, a_2 \,\leq\, a_3 \,\leq\, \cdots,</math>
there exists a positive [[integer]] ''n'' such that
:<math>a_n = a_{n+1} = a_{n+2} = \cdots.</math>
Similarly, ''P'' is said to satisfy the '''descending chain condition''' (DCC) if every strictly descending sequence of elements eventually terminates, that is, there is no [[infinite descending chain]]. Equivalently every descending sequence
:<math>\cdots \,\leq\, a_3 \,\leq\, a_2 \,\leq\, a_1</math>
of elements of ''P'', eventually stabilizes.
 
=== 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.
 
* A [[total order|totally ordered set]] that satisfies the descending chain condition is called a [[well-order|well-ordered set]].
 
== See also ==
* [[Artinian]]
* [[Noetherian]]
* [[Krull dimension]]
* [[Ascending chain condition for principal ideals]]
* [[Maximal condition on congruences]]
 
== Notes ==
<references/>
 
== References ==
* [[M. F. Atiyah|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
 
{{DEFAULTSORT:Ascending Chain Condition}}
[[Category:Commutative algebra]]
[[Category:Order theory]]
[[Category:Wellfoundedness]]

Latest revision as of 16:13, 10 September 2013

In mathematics, 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

a1a2a3,

there exists a positive integer n such that

an=an+1=an+2=.

Similarly, P is said to satisfy the descending chain condition (DCC) if every strictly descending sequence of elements eventually terminates, that is, there is no infinite descending chain. Equivalently every descending sequence

a3a2a1

of elements of P, eventually stabilizes.

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