Ascending chain condition

Definition

A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence : a_1 of elements of P exists. Equivalently, every weakly ascending sequence : a_1 \leq a_2 \leq a_3 \leq \cdots, of elements of P eventually stabilizes, meaning that 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 strictly descending chain of elements of P. Equivalently, every weakly descending sequence : a_1 \geq a_2 \geq a_3 \geq \cdots of elements of P eventually stabilizes.

Comments

• Assuming the axiom of dependent choice, the descending chain condition on (possibly infinite) poset 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). A totally ordered set that is well-founded is a well-ordered set.
• Similarly, the ascending chain condition is equivalent to P being converse well-founded (again, assuming dependent choice): every nonempty subset of P has a maximal element (the maximal condition or maximum condition).
• Every finite poset satisfies both the ascending and descending chain conditions, and thus is both well-founded and converse well-founded.

Example

Consider the ring : \mathbb{Z} = {\dots, -3, -2, -1, 0, 1, 2, 3, \dots} of integers. Each ideal of \mathbb{Z} consists of all multiples of some number n. For example, the ideal : I = {\dots, -18, -12, -6, 0, 6, 12, 18, \dots} consists of all multiples of 6. Let : J = {\dots, -6, -4, -2, 0, 2, 4, 6, \dots} be the ideal consisting of all multiples of 2. The ideal I is contained inside the ideal J, since every multiple of 6 is also a multiple of 2. In turn, the ideal J is contained in the ideal \mathbb{Z}, since every multiple of 2 is a multiple of 1. However, at this point there is no larger ideal; we have "topped out" at \mathbb{Z}.

In general, if I_1, I_2, I_3, \dots are ideals of \mathbb{Z} such that I_1 is contained in I_2, I_2 is contained in I_3, and so on, then there is some n for which all I_n = I_{n+1} = I_{n+2} = \cdots. That is, after some point all the ideals are equal to each other. Therefore, the ideals of \mathbb{Z} satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence \mathbb{Z} is a Noetherian ring.

Notes

Citations

References

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