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,[lower-alpha 1] 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 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.

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?".