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 every strictly ascending sequence of elements eventually terminates. Equivalently, given any sequence

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

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 every strictly descending sequence of elements eventually terminates, that is, there is no infinite descending chain. Equivalently every descending sequence

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

of elements of P, eventually stabilizes.

Comments

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).
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).
Trivially every finite poset satisfies both ACC and DCC.
A totally ordered set that satisfies the descending chain condition is called a well-ordered set.

Notes

↑ Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.
↑ Fraleigh & Katz (1967), p. 366, Lemma 7.1
↑ Jacobson (2009), p. 142 and 147

References

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

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Ascending chain condition

Definition

Comments

See also

Notes

References