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 Lasker–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

### Other articles related to "noetherian ring, noetherian, ring":

**Noetherian Ring**- Properties

... If R is a

**Noetherian ring**, then R is

**Noetherian**by the Hilbert basis theorem ... Also, R], the power series

**ring**is a

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... chain conditions, and usually if they satisfy an ascending chain condition, they are called

**Noetherian**in her honor ... By definition, a

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**Noetherian**module is a module in which every strictly ascending chain of submodules breaks off after a finite number ...

... modules states every submodule of a finitely generated module over a

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**Noetherian ring**is contained in a finite product of coprimary modules ...

... It is possible, even for a commutative local

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... named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a

**Noetherian ring**... Formally, if R is a

**Noetherian ring**and I is a principal, proper ideal of R, then I has height at most one ... This says that if R is a

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### Famous quotes containing the word ring:

“I saw Eternity the other night,

Like a great *ring* of pure and endless light,”

—Henry Vaughan (1622–1695)