 Ascending chain condition

The ascending chain condition requires every chainof ideals to stabilize. A ring is Noetherian exactly when it satisfies this condition, equivalently when every ideal is finitely generated.

The Ascending Chain Condition (ACC) is a property related to partially ordered sets (posets) and certain algebraic structures in mathematics, particularly in order theory and abstract algebra. **Definition:** A partially ordered set satisfies the Ascending Chain Condition if every ascending chain of elements eventually stabilizes.