Inmathematics, theascending chain condition (ACC) anddescending chain condition (DCC) are finiteness properties satisfied by somealgebraic structures, most importantlyideals in certaincommutative rings.^[1][2][3] These conditions played an important role in the development of the structure theory of commutative rings in the works ofDavid Hilbert,Emmy Noether, andEmil Artin.The conditions themselves can be stated in an abstract form, so that they make sense for anypartially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

Apartially ordered set (poset)P is said to satisfy theascending chain condition (ACC) if no infinite strictly ascending sequence

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

${\displaystyle a_1\le a_2\le a_3\le \cdots ,}$

of elements ofP eventually stabilizes, meaning that there exists a positive integern such that

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

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

${\displaystyle a_1\ge a_2\ge a_3\ge \cdots }$

of elements ofP eventually stabilizes.

• Assuming theaxiom of dependent choice, the descending chain condition on (possibly infinite) posetP is equivalent toP beingwell-founded: every nonempty subset ofP has a minimal element (also called theminimal condition orminimum condition). Atotally ordered set that is well-founded is awell-ordered set.
• Similarly, the ascending chain condition is equivalent toP being converse well-founded (again, assuming dependent choice): every nonempty subset ofP has a maximal element (themaximal condition ormaximum 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 aNoetherian 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?".

• Commutative algebra
• Order theory
• Wellfoundedness

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