Ascending chain condition

A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence

${\displaystyle a_{1}<a_{2}<a_{3}<\cdots }$ 

of elements of P exists.^[4] Equivalently,^[a] every weakly ascending sequence

${\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq \cdots ,}$ 

of elements of P eventually stabilizes, meaning that there exists a positive integer n such that

${\displaystyle a_{n}=a_{n+1}=a_{n+2}=\cdots .}$ 

Similarly, P is said to satisfy the descending chain condition (DCC) if there is no infinite strictly descending chain of elements of P.^[4] Equivalently, every weakly descending sequence

${\displaystyle a_{1}\geq a_{2}\geq a_{3}\geq \cdots }$ 

of elements of P eventually stabilizes.

• 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 totally ordered set that is well-founded is a 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.

Consider the ring

${\displaystyle \mathbb {Z} =\{\dots ,-3,-2,-1,0,1,2,3,\dots \}}$ 

of integers. Each ideal of ${\displaystyle \mathbb {Z} }$  consists of all multiples of some number ${\displaystyle n}$ . For example, the ideal

${\displaystyle I=\{\dots ,-18,-12,-6,0,6,12,18,\dots \}}$ 

consists of all multiples of ${\displaystyle 6}$ . Let

${\displaystyle J=\{\dots ,-6,-4,-2,0,2,4,6,\dots \}}$ 

be the ideal consisting of all multiples of ${\displaystyle 2}$ . The ideal ${\displaystyle I}$  is contained inside the ideal ${\displaystyle J}$ , since every multiple of ${\displaystyle 6}$  is also a multiple of ${\displaystyle 2}$ . In turn, the ideal ${\displaystyle J}$  is contained in the ideal ${\displaystyle \mathbb {Z} }$ , since every multiple of ${\displaystyle 2}$  is a multiple of ${\displaystyle 1}$ . However, at this point there is no larger ideal; we have "topped out" at ${\displaystyle \mathbb {Z} }$ .

In general, if ${\displaystyle I_{1},I_{2},I_{3},\dots }$  are ideals of ${\displaystyle \mathbb {Z} }$  such that ${\displaystyle I_{1}}$  is contained in ${\displaystyle I_{2}}$ , ${\displaystyle I_{2}}$  is contained in ${\displaystyle I_{3}}$ , and so on, then there is some ${\displaystyle n}$  for which all ${\displaystyle I_{n}=I_{n+1}=I_{n+2}=\cdots }$ . That is, after some point all the ideals are equal to each other. Therefore, the ideals of ${\displaystyle \mathbb {Z} }$  satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence ${\displaystyle \mathbb {Z} }$  is a Noetherian ring.

• Artinian
• Ascending chain condition for principal ideals
• Krull dimension
• Maximal condition on congruences
• Noetherian

• "Is the equivalence of the ascending chain condition and the maximum condition equivalent to the axiom of dependent choice?".