Ascending chain condition

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 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.

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 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.

Example

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.

See also

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

Notes

1. 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. Notice the proof does not use the full force of the axiom of choice.^{[ clarification needed ]}

Citations

1. Hazewinkel, Gubareni & Kirichenko 2004, p. 6, Prop. 1.1.4
2. Fraleigh & Katz 1967, p. 366, Lemma 7.1
3. Jacobson 2009, pp. 142, 147
4. Hazewinkel, p. 580

References

• Atiyah, M. F.; MacDonald, I. G. (1969), Introduction to Commutative Algebra , Perseus Books, ISBN   0-201-00361-9
• Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2004), Algebras, rings and modules, Kluwer Academic Publishers, ISBN   1-4020-2690-0
• Hazewinkel, Michiel. Encyclopaedia of Mathematics. Kluwer. ISBN   1-55608-010-7.
• Fraleigh, John B.; Katz, Victor J. (1967), A first course in abstract algebra (5th ed.), Addison-Wesley Publishing Company, ISBN   0-201-53467-3
• Jacobson, Nathan (2009), Basic Algebra I, Dover, ISBN   978-0-486-47189-1

External links

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