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 Nöther, 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.
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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 am = an 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).
Comments
- The ascending chain condition on P is equivalent to the maximal condition: every nonempty subset of P has a maximal element.
- Similarly, the descending chain condition is equivalent to the minimal condition: every nonempty subset of P has a minimal element.
- Every finite poset satisfies both ACC and DCC.
- A totally ordered set that satisfies the descending chain condition is called a well-ordered set.
See also
References
- Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9
