In mathematics, more specifically in the area of modern algebra known as ring theory, a **Noetherian ring**, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:

there exists a positive integer *n* such that:

There are other equivalent formulations of the definition of a **Noetherian ring** and these are outlined later in the article.

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Laskerâ€“Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on *prime ideals*. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

Read more about Noetherian Ring: Introduction, Characterizations, Hilbert's Basis Theorem, Primary Decomposition, Uses, Examples, Properties

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—Harriet Beecher Stowe (1811–1896)