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. 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 weakly ascending 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 of P eventually terminates, that is, there is no infinite descending chain. Equivalently, every weakly descending sequence

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

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 a well-ordered set (assuming the axiom of dependent choice).

See also

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

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