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.
To prove a property of all elements of a partially ordered set satisfying the ascending chain condition, assume a counterexample exists and take a maximal counterexample. All strictly larger elements satisfy the property, allowing an inductive contradiction. Applied to ideals, this often replaces induction on a numerical size.
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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.