Ascending chain condition

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The ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by certain algebraic structures, most importantly, ideals in a commutative ring. 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.

Contents 1 Definition 1.1 Comments 2 See also 3 References

[edit] Definition

A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if every ascending chain a₁ ≤ a₂ ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that a_m = a_n for all m > n. Similarly, P is said to satisfy the descending chain condition (DCC) if every descending chain a₁ ≥ a₂ ≥ ... of elements of P is eventually stationary (that is, there is no infinite descending chain).

[edit] 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).

• Every finite poset satisfies both ACC and DCC.

• A totally ordered set that satisfies the descending chain condition is called a well-ordered set.

[edit] See also

• Noetherian
• Artinian
• Krull dimension

[edit] References

• Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9