* Noetherian ring

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

If is a left-( respectively right -) Noetherian ring, then the polynomial ring is also a left-( respectively right -) Noetherian ring.

It can be achieved in many ways, such as requiring the ring to be right Noetherian.

The prime ideals of the ring of integers are the ideals ( 0 ), ( 2 ), ( 3 ), ( 5 ), ( 7 ), ( 11 ), … The fundamental theorem of arithmetic generalizes to the Lasker – Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

The Krull dimension need not be finite even for a Noetherian ring.

Nagata also gave an example of a Noetherian ring where not every chain can be extended to a maximal chain.

In general, if R is a Noetherian ring of dimension d, then the dimension of R is d + 1.

In general, a Noetherian ring is Artinian if and only if its Krull dimension is 0.

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 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.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one ; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals.

In commutative algebra, 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.

* A Noetherian integral domain is a unique factorization domain if and only if every height 1 prime ideal is principal.

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.

This fact follows from the famous Hilbert's basis theorem named after mathematician David Hilbert ; the theorem asserts that if R is any Noetherian ring ( such as, for instance, ), R is also a Noetherian ring.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

While the ideal structure of becomes considerably more complex as n increases, the rings in question still remain Noetherian, and any theorem about that can be proven using only the fact that is Noetherian, can be proven for.

* If R is a Noetherian ring, then R is Noetherian by the Hilbert basis theorem.

* if R is Noetherian, then so is R < nowiki ></ nowiki > X < nowiki ></ nowiki >; this is a version of the Hilbert basis theorem

* The Hopkins – Levitzki theorem gives necessary and sufficient conditions for a Noetherian ring to be an Artinian ring.

Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length ; there is also analogous theorem for coherent sheaves when the algebra is Noetherian.

Specifically, a consequence of the Akizuki – Hopkins – Levitzki theorem is that a left ( right ) Artinian ring is automatically a left ( right ) Noetherian ring.

A Noetherian ring is Cohen – Macaulay if and only if the unmixedness theorem holds for it.

In mathematics, the Lasker – Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection of finitely many primary ideals ( which are related to, but not quite the same as, powers of prime ideals ).

The Lasker – Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings.

# Every principal ideal domain is Noetherian.

In fact, it is possible to give a proof that is a Noetherian ring without appealing to its order structure and this proof applies more generally to principal ideal rings ( i. e., rings in which every ideal is generated by a single element ).

Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various " natural rings ".

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.

Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:

* Any principal ideal domain, such as the integers, is Noetherian since every ideal is generated by a single element.

* A Dedekind domain ( e. g., rings of integers ) is Noetherian since every ideal is generated by at most two elements.

* The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I < sub > n </ sub > be the ideal of all continuous functions f such that f ( x ) = 0 for all x ≥ n. The sequence of ideals I < sub > 0 </ sub >, I < sub > 1 </ sub >, I < sub > 2 </ sub >, etc., is an ascending chain that does not terminate.

* If R is a Noetherian ring and I is a two-sided ideal, then the factor ring R / I is also Noetherian.

If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.

Another illustration of the delicate / robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property -- a Noetherian domain is Dedekind iff for every maximal ideal of the localization is a Dedekind ring.

( DD2 ) is Noetherian, and the localization at each maximal ideal is a Discrete Valuation Ring.