Editing Ascending chain condition

{{Short description|Condition in commutative algebra}}
In [[mathematics]], the '''ascending chain condition''' ('''ACC''') and '''descending chain condition''' ('''DCC''') are finiteness properties satisfied by some [[algebraic structure]]s, most importantly [[Ideal (ring theory)|ideal]]s in certain [[commutative ring]]s.{{sfn|Hazewinkel|Gubareni|Kirichenko|2004|loc=p. 6, Prop. 1.1.4|ps=none}}{{sfn|Fraleigh|Katz|1967|loc=p. 366, Lemma 7.1|ps=none}}{{sfn|Jacobson|2009|pp=142,147|ps=none}} 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 no infinite strictly ascending sequence 
: <math>a_1 < a_2 < a_3 < \cdots</math>
of elements of ''P'' exists.{{sfn|Hazewinkel|p=580|ps=none}} 
Equivalently,{{efn|Proof: first, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ascending sequence that does not stabilize; then clearly it contains a strictly increasing (necessarily infinite) subsequence.}} every weakly ascending sequence
: <math>a_1 \leq a_2 \leq a_3 \leq \cdots,</math>
of elements of ''P'' eventually stabilizes, meaning that 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 there is no infinite strictly descending chain of elements of ''P''.{{sfn|Hazewinkel|p=580|ps=none}} Equivalently, every weakly descending sequence
: <math>a_1 \geq a_2 \geq a_3 \geq \cdots</math>
of elements of ''P'' eventually stabilizes.

=== Comments ===
* Assuming the [[axiom of dependent choice]], the descending chain condition on (possibly infinite) poset ''P'' is equivalent to ''P'' being [[well-founded]]: every nonempty subset of ''P'' has a minimal element (also called the '''minimal condition''' or '''minimum condition'''). A [[total order|totally ordered set]] that is well-founded is a [[well-order|well-ordered set]].
* Similarly, the ascending chain condition is equivalent to ''P'' being converse well-founded (again, assuming dependent choice): every nonempty subset of ''P'' has a maximal element (the '''maximal condition''' or '''maximum condition''').
* Every finite poset satisfies both the ascending and descending chain conditions, and thus is both well-founded and converse well-founded.

== Example ==
Consider the ring
: <math>\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}</math>
of integers. Each ideal of <math>\mathbb{Z}</math> consists of all multiples of some number <math>n</math>. For example, the ideal
: <math>I = \{\dots, -18, -12, -6, 0, 6, 12, 18, \dots\}</math>
consists of all multiples of <math>6</math>. Let
: <math>J = \{\dots, -6, -4, -2, 0, 2, 4, 6, \dots\}</math>
be the ideal consisting of all multiples of <math>2</math>. The ideal <math>I</math> is contained inside the ideal <math>J</math>, since every multiple of <math>6</math> is also a multiple of <math>2</math>. In turn, the ideal <math>J</math> is contained in the ideal <math>\mathbb{Z}</math>, since every multiple of <math>2</math> is a multiple of <math>1</math>. However, at this point there is no larger ideal; we have "topped out" at <math>\mathbb{Z}</math>.

In general, if <math>I_1, I_2, I_3, \dots</math> are ideals of <math>\mathbb{Z}</math> such that <math>I_1</math> is contained in <math>I_2</math>, <math>I_2</math> is contained in <math>I_3</math>, and so on, then there is some <math>n</math> for which all <math>I_n = I_{n+1} = I_{n+2} = \cdots</math>. That is, after some point all the ideals are equal to each other. Therefore, the ideals of <math>\mathbb{Z}</math> satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence <math>\mathbb{Z}</math> is a [[Noetherian ring]].

== See also ==
* [[Artinian (disambiguation)|Artinian]]
* [[Ascending chain condition for principal ideals]]
* [[Krull dimension]]
* [[Maximal condition on congruences]]
* [[Noetherian]]

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation
  | last1=Atiyah | first1=M. F. | author1-link=M. F. Atiyah
  | last2=MacDonald | first2=I. G.
  | title=[[Introduction to Commutative Algebra]]
  | publisher=Perseus Books
  | year=1969
  | isbn=0-201-00361-9
  }}
* {{citation
  | last1=Hazewinkel | first1=Michiel | author1-link=Michiel Hazewinkel
  | last2=Gubareni | first2=Nadiya
  | last3=Kirichenko | first3=V. V.
  | title=Algebras, rings and modules
  | publisher=[[Kluwer Academic Publishers]]
  | year=2004
  | isbn=1-4020-2690-0
  }}
* {{cite book
  | last = Hazewinkel | first = Michiel
  | title=Encyclopaedia of Mathematics
  | publisher=Kluwer
  | isbn=1-55608-010-7
  }}
* {{citation
  | last1=Fraleigh | first1=John B.
  | last2=Katz | first2=Victor J.
  | title=A first course in abstract algebra
  | publisher=Addison-Wesley Publishing Company
  | edition=5th
  | year=1967
  | isbn=0-201-53467-3
  }}
* {{citation
  | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson
  | title=Basic Algebra I
  | publisher=Dover
  | year=2009 
  | isbn=978-0-486-47189-1
  }}
{{refend}}

== External links ==
* {{cite web |title=Is the equivalence of the ascending chain condition and the maximum condition equivalent to the axiom of dependent choice? |url=https://math.stackexchange.com/q/1746921 }}

{{DEFAULTSORT:Ascending Chain Condition}}
[[Category:Commutative algebra]]
[[Category:Order theory]]
[[Category:Wellfoundedness]]