In
mathematics, a Noetherian ring is a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
that satisfies the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These ...
on left and right
ideals
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence
of left (or right) ideals has a largest element; that is, there exists an such that:
Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is
finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.
Noetherian rings are fundamental in both
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
and
noncommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
ring theory since many rings that are encountered in mathematics are Noetherian (in particular the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
,
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
s, and
rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the
Lasker–Noether theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many '' primary ideals'' (which are relat ...
and the
Krull intersection theorem In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
).
Noetherian rings are named after
Emmy Noether
Amalie Emmy Noether Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
, but the importance of the concept was recognized earlier by
David Hilbert, with the proof of
Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and
Hilbert's syzygy theorem.
Characterizations
For
noncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s, it is necessary to distinguish between three very similar concepts:
* A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals.
* A ring is right-Noetherian if it satisfies the ascending chain condition on right ideals.
* A ring is Noetherian if it is both left- and right-Noetherian.
For
commutative rings, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa.
There are other, equivalent, definitions for a ring ''R'' to be left-Noetherian:
* Every left ideal ''I'' in ''R'' is
finitely generated, i.e. there exist elements
in ''I'' such that
.
[Lam (2001), p. 19]
* Every
non-empty set of left ideals of ''R'',
partially ordered
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
by inclusion, has a
maximal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is def ...
.
Similar results hold for right-Noetherian rings.
The following condition is also an equivalent condition for a ring ''R'' to be left-Noetherian and it is
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
's original formulation:
*Given a sequence
of elements in ''R'', there exists an integer
such that each
is a finite linear combination
with coefficients
in ''R''.
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. However, it is not enough to ask that all the maximal ideals are finitely generated, as there is a non-Noetherian
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
whose maximal ideal is principal (see a counterexample to Krull’s intersection theorem at
Local ring#Commutative case.)
Properties
* If ''R'' is a Noetherian ring, then the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...