A Noetherian ring is a specific type of ring in algebra that satisfies a property related to the concept of ideal containment. A ring \( R \) is called Noetherian if it satisfies any of the following equivalent conditions: 1. **Ascending Chain Condition on Ideals (ACC)**: Every ascending chain of ideals in \( R \) stabilizes.
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A commutative ring is Noetherian if each of its ideals has a finite generating set. Equivalently, every increasing chain of ideals eventually stops growing. For the equivalence, the union of an increasing chain is an ideal; its finitely many generators already lie in one member. Conversely, repeatedly adjoining an element outside the generated ideal would produce a nonterminating chain.
The integers are a Noetherian ring, since each ideal is principal. The Hilbert basis theorem then shows that is Noetherian. The infinitely generated polynomial ring is not: never stabilizes. The condition ensures that finitely many equations suffice to describe each ideal, even when its elements form an infinite set.[ref]