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[edit]

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

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

of elements of P exists.^[4] Equivalently,^{[note 1]} given any weakly ascending sequence

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

of elements of P there exists a positive integer n such that

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

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

of elements of P eventually stabilizes.

Comments[edit]

The descending chain condition on 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).^{[clarification needed]}
Similarly, the ascending chain condition is equivalent to P being converse well-founded: every nonempty subset of P has a maximal element (the maximal condition or maximum condition).
Trivially every finite poset satisfies both ACC and DCC.
A totally ordered set that satisfies the descending chain condition is a well-ordered set (assuming the axiom of dependent choice).

Notes[edit]

^ 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]}

^ Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.
^ Fraleigh & Katz (1967), p. 366, Lemma 7.1
^ Jacobson (2009), p. 142 and 147
^ ^a ^b Hazewinkel, Michiel. Encyclopaedia of Mathematics. Kluwer. p. 580. ISBN 1-55608-010-7.

References[edit]

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

External links[edit]

https://math.stackexchange.com/questions/1746921/is-the-equivalence-of-the-ascending-chain-condition-and-the-maximum-condition-eq

[5] 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]}

[1] Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.

[2] Fraleigh & Katz (1967), p. 366, Lemma 7.1

[3] Jacobson (2009), p. 142 and 147

[Hazewinkel-4] Hazewinkel, Michiel. Encyclopaedia of Mathematics. Kluwer. p. 580. ISBN 1-55608-010-7.

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[note 1]

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Tuesday, March 10, 2020