Noetherian Ring

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 RingIntroduction, 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 Noetherian ring ... If R is a Noetherian ring and I is a two-sided ideal, then the factor ring R/I is also Noetherian ...
Emmy Noether - Contributions To Mathematics and Physics - Second Epoch (1920–26) - Ascending and Descending Chain Conditions
... chain conditions, and usually if they satisfy an ascending chain condition, they are called Noetherian in her honor ... By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is ... A Noetherian module is a module in which every strictly ascending chain of submodules breaks off after a finite number ...
Lasker–Noether Theorem - Statement
... modules states every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules ... of ideals it states that every ideal of a Noetherian ring is a finite intersection of primary ideals ... is every finitely generated module over a Noetherian ring is contained in a finite product of coprimary modules ...
Associated Prime - Properties
... It is possible, even for a commutative local ring, that the set of associated primes of a finitely generated module is empty ... However, in any ring satisfying the ascending chain condition on ideals (for example, any right or left Noetherian ring) every nonzero module has at least one associated prime ... For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the spectrum ...
Krull's Principal Ideal Theorem
... 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 Noetherian ring and I is a proper ideal generated by n elements of R, then I has height at most n ...

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