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

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

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...

or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric...

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

lattice (or its right counterpart) satisfies the ascending chain condition or descending chain condition. Denote the lattice of left annihilator ideals...

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

well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewriting systems, a Noetherian relation is...

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

have a greatest element. If P {\displaystyle P} satisfies the ascending chain condition, a subset S {\displaystyle S} of P {\displaystyle P} has a greatest...

is principal (i.e., A is a Bézout domain) and A satisfies the ascending chain condition on principal ideals. A admits a Dedekind–Hasse norm. Any Euclidean...

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

(="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states...

Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R (with 1)...

series, while the upper central series is an ascending series. A group that satisfies the ascending chain condition (ACC) on subgroups is called a Noetherian...

element; see example 3. If P {\displaystyle P} satisfies the ascending chain condition, a subset S {\displaystyle S} of P {\displaystyle P} has a greatest...

module is a module that satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after...

"finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states...

numbers. Three conditions were required: an ascending chain condition, a dimension condition, and the condition that the ring be integrally closed. |} In...

nonzero prime ideal of A contains a prime element. A satisfies ascending chain condition on principal ideals (ACCP), and the localization S−1A is a UFD...

partial order (or more generally a partial order satisfying the ascending chain condition) all lower sets have this form. The union of any two lower sets...

differential polynomials over K {\displaystyle K} satisfy the ascending chain condition on radical differential ideals. This Ritt’s theorem is implied...

satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian). GCD domains appear in the following chain of class inclusions:...

ring Hilbert's basis theorem Artinian ring Ascending chain condition (ACC) and descending chain condition (DCC) Fractional ideal Ideal class group Radical...

recast many earlier results in terms of an ascending chain condition, now known as the Noetherian condition. Another important milestone was the work of...

this is equivalent to saying that the ascending chain condition holds: there is no infinite strictly ascending chain of congruences on S. Every ideal I of...

is because all UFDs satisfy the ascending chain condition on principal ideals, but there is an infinite ascending chain of principal ideals ⋯ ⊊ ( x − j...