ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence a 1 < a 2 < a 3 < ⋯ {\displaystyle...

inclusion. The ascending chain condition on principal ideals (abbreviated to ACCP) is satisfied if there is no infinite strictly ascending chain of principal...

the ascending chain condition if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if...

the descending chain condition. Similarly, the ascending chain condition means that every ascending chain eventually stabilizes. For example, a Noetherian...

objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences...

Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for...

the chain), so there cannot be any infinite strictly ascending chain of principal ideals of R. That condition, called the ascending chain condition on...

satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements...

abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered...

Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition. Specifically, a consequence...