Noetherian ring
Encyclopedia
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 ideal
s 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 integer
s 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 ideal
s. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension
.
denote the ring
of integers; that is, let
be the set of integers equipped with its natural operations of addition and multiplication. An ideal in 
is a subset
, I, of
that is closed under subtraction (i.e., if 
, 
), and closed under "inside-outside multiplication" (i.e., if r is any integer, not necessarily in I, and i is any element of I, 
). In fact, in the general case of a ring
, these two requirements define the notion of an ideal in a ring. It is a fact that the ring
is a principal ideal ring
; that is, for any ideal I in
, there exists an integer n in I such that every element of I is a multiple of n. Conversely, the set of all multiples of an arbitrary integer n is necessarily an ideal, and is usually denoted by (n).
Although there are many (equivalent) formulations of what it means for a ring, R, to be Noetherian, one formulation dictates that any ascending chain of ideals in R terminates. That is, if:
is an ascending chain of ideals, then there exists a positive integer n such that
For instance, if I and J are ideals in
, there exists integers n and m such that I=(n) and J=(m) (i.e., every integer in I is a multiple of n and every integer in J is a multiple of m). In this case, 
if and only if every element of I is an element of J, or equivalently, if every multiple of the integer n is a multiple of m. In other words, 
if and only if m divides n (or m is a factor of n). Furthermore, the inclusion 
is proper if and only if m is a proper divisor of n (i.e., 
with k not equal to either 1 or −1).
Thus, if:
is an ascending chain of ideals in
and Ij=(nj) for all j and integers nj, nj+1 divides nj for all j. If each inclusion is proper (that is, if the chain does not terminate), n2 would be a proper divisor of n1, n3 would be a proper divisor of n2 etc. In particular, 
which is impossible since there can only be finitely many positive integers strictly less than n1. Consequently, 
is a Noetherian ring.
For this reason, the notion of a Noetherian ring generalizes such rings as
. The fundamental property of 
used in the proof above is that there cannot be a chain of positive integers where each integer in the chain is strictly less than its predecessor; in other words, the ring of integers is not "too large" since it cannot sustain such a "large chain". This is typical in the theory of Noetherian rings; often, to prove a result about Noetherian rings, one appeals to the fact that the rings in question are not "too large". More formally, one assumes that the conclusion of the result is false and exhibits an ascending chain that does not terminate thus contradicting the fact that the ring is "not too large", and establishing that the conclusion must, in fact, be true.
While the proof that
is a Noetherian ring uses the order structure of 
, typical proofs in ring theory in general do not assume such additional structure on the ring. 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).
Although the ring
is a Noetherian ring, the theory of Noetherian rings extends far beyond just this ring. For example, let 
denote the polynomial ring
in one indeterminant over
. More specifically, let 
be the set of all polynomials with integer coefficients (such a polynomial is also referred to as a polynomial over 
), with addition and multiplication defined to be natural polynomial addition and multiplication. Under these operations 
becomes a ring. More generally, if R is any ring, the set of all polynomials with coefficients in R can be equipped with the structure of a ring and is denoted by R[X].
Although every ideal in
is simply the set of multiples of a certain integer n, the ideal structure of 
is slightly more complicated; there are ideals that may not be expressed as the set of multiples of a given polynomial. Put differently, 
is not a principal ideal ring. However, it is a Noetherian ring. 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[X] is also a Noetherian ring. In fact, by the principle of mathematical induction, Hilbert's basis theorem establishes that 
, the ring of all polynomials in n variables with coefficients in 
, is a Noetherian ring.
Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various "natural rings". 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 
.
s, it is necessary to distinguish between three very similar concepts:
For commutative ring
s, 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:
Similar results hold for right-Noetherian rings.
It is also known that for a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to I. S. Cohen.)
proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R[X]. Formally,
Theorem
If R is a Noetherian ring, then R[X] is a Noetherian ring.
Corollary
If R is a Noetherian ring, then
is a Noetherian ring.
For a proof of this result, see the corresponding section on the Hilbert's basis theorem page. Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n variables is generally a geometric object (such as a curve or a surface) in n-space).
of integers, an arbitrary ideal is of the form (n) for some integer n (where (n) denotes the set of all integer multiples of n). If n is non-zero, and is neither 1 nor −1, by the fundamental theorem of arithmetic
, there exist primes pi, and positive integers ei, with
. In this case, the ideal (n) may be written as the intersection of the ideals (piei); that is, 
. This is referred to as a primary decomposition of the ideal (n).
In general, an ideal Q of a ring is said to be primary
if Q is proper and whenever
, either 
or 
for some positive integer n. In 
, the primary ideals are precisely the ideals of the form (pe) where p is prime and e is a positive integer. Thus, a primary decomposition of (n) corresponds to representing (n) as the intersection of finitely many primary ideals.
Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor −1 also asserts uniqueness of the representation
for pi prime and ei positive, a primary decomposition of (n) is essentially unique.
For all of the above reasons, the following theorem, referred to as the Lasker–Noether theorem
, may be seen as a certain generalization of the fundamental theorem of arithmetic:
Theorem
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:
with Qi primary for all i and
for 
. Furthermore, if:
is decomposition of I with
for 
, and both decompositions of I are irredundant (meaning that no proper subset of either 
or 
yields an intersection equal to I), 
and (after possibly renumbering the Qi's) 
for all i.
For any primary decomposition of I, the set of all radicals, that is, the set
remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module R/I; that is, the set of all annihilators
of R/I (viewed as a module over R) that are prime.
and in areas that make heavy use of rings, such as algebraic geometry
. The reason behind this is that the Noetherian property is in some sense the ring-theoretic analogue of finiteness. For example, the fact that polynomial rings over a field are Noetherian allows one to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions.
Krull's principal ideal theorem
is an important property of Noetherian rings. It 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. This early result was the first to suggest that Noetherian rings possessed a deep theory of dimension
.
Rings that are not Noetherian tend to be (in some sense) very large. Here are two examples of non-Noetherian rings:
However, a non-Noetherian ring can be a subring of a Noetherian ring:
Indeed, there are rings that are left Noetherian, but not right Noetherian, so that one must be careful in measuring the "size" of a ring this way.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically in the area of modern algebra known as ring theory
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, a Noetherian ring, named after Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
, is a ring in which every non-empty set of ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition
Ascending chain condition
The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
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 integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s and the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem
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 written as an intersection of finitely many primary ideals...
, the Krull intersection theorem, and the Hilbert's basis theorem
Hilbert's basis theorem
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. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
.
Introduction
Let denote the ringRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of integers; that is, let
be the set of integers equipped with its natural operations of addition and multiplication. An ideal in is a subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
, I, of
that is closed under subtraction (i.e., if , ), and closed under "inside-outside multiplication" (i.e., if r is any integer, not necessarily in I, and i is any element of I, ). In fact, in the general case of a ringRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, these two requirements define the notion of an ideal in a ring. It is a fact that the ring
is a principal ideal ringPrincipal ideal ring
In mathematics, a principal right ideal ring is a ring R in which every right ideal is of the form xR for some element x of R...
; that is, for any ideal I in
, there exists an integer n in I such that every element of I is a multiple of n. Conversely, the set of all multiples of an arbitrary integer n is necessarily an ideal, and is usually denoted by (n).Although there are many (equivalent) formulations of what it means for a ring, R, to be Noetherian, one formulation dictates that any ascending chain of ideals in R terminates. That is, if:
is an ascending chain of ideals, then there exists a positive integer n such that
For instance, if I and J are ideals in
, there exists integers n and m such that I=(n) and J=(m) (i.e., every integer in I is a multiple of n and every integer in J is a multiple of m). In this case, if and only if every element of I is an element of J, or equivalently, if every multiple of the integer n is a multiple of m. In other words, if and only if m divides n (or m is a factor of n). Furthermore, the inclusion is proper if and only if m is a proper divisor of n (i.e., with k not equal to either 1 or −1).Thus, if:
is an ascending chain of ideals in
and Ij=(nj) for all j and integers nj, nj+1 divides nj for all j. If each inclusion is proper (that is, if the chain does not terminate), n2 would be a proper divisor of n1, n3 would be a proper divisor of n2 etc. In particular, which is impossible since there can only be finitely many positive integers strictly less than n1. Consequently, is a Noetherian ring.For this reason, the notion of a Noetherian ring generalizes such rings as
. The fundamental property of used in the proof above is that there cannot be a chain of positive integers where each integer in the chain is strictly less than its predecessor; in other words, the ring of integers is not "too large" since it cannot sustain such a "large chain". This is typical in the theory of Noetherian rings; often, to prove a result about Noetherian rings, one appeals to the fact that the rings in question are not "too large". More formally, one assumes that the conclusion of the result is false and exhibits an ascending chain that does not terminate thus contradicting the fact that the ring is "not too large", and establishing that the conclusion must, in fact, be true.While the proof that
is a Noetherian ring uses the order structure of , typical proofs in ring theory in general do not assume such additional structure on the ring. 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).Although the ring
is a Noetherian ring, the theory of Noetherian rings extends far beyond just this ring. For example, let denote the polynomial ringPolynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
in one indeterminant over
. More specifically, let be the set of all polynomials with integer coefficients (such a polynomial is also referred to as a polynomial over ), with addition and multiplication defined to be natural polynomial addition and multiplication. Under these operations becomes a ring. More generally, if R is any ring, the set of all polynomials with coefficients in R can be equipped with the structure of a ring and is denoted by R[X].Although every ideal in
is simply the set of multiples of a certain integer n, the ideal structure of is slightly more complicated; there are ideals that may not be expressed as the set of multiples of a given polynomial. Put differently, is not a principal ideal ring. However, it is a Noetherian ring. This fact follows from the famous Hilbert's basis theoremHilbert's basis theorem
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. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
named after mathematician David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
; the theorem asserts that if R is any Noetherian ring (such as, for instance,
), R[X] is also a Noetherian ring. In fact, by the principle of mathematical induction, Hilbert's basis theorem establishes that , the ring of all polynomials in n variables with coefficients in , is a Noetherian ring.Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various "natural rings". 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 .Characterizations
For noncommutative ringNoncommutative ring
In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.Noncommutative rings are ubiquitous in mathematics, and occur...
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 ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s, 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 a1, ..., an in I such that I = Ra1 + ... + Ran.
- Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal elementMaximal elementIn mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually...
with respect to set inclusionSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
.
Similar results hold for right-Noetherian rings.
It is also known that for a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to I. S. Cohen.)
Hilbert's basis theorem
If R is a ring, let R[X] denote the ring of polynomials in the indeterminant X over R. HilbertDavid Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R[X]. Formally,
Theorem
If R is a Noetherian ring, then R[X] is a Noetherian ring.
Corollary
If R is a Noetherian ring, then
is a Noetherian ring.For a proof of this result, see the corresponding section on the Hilbert's basis theorem page. Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n variables is generally a geometric object (such as a curve or a surface) in n-space).
Primary decomposition
In the ring of integers, an arbitrary ideal is of the form (n) for some integer n (where (n) denotes the set of all integer multiples of n). If n is non-zero, and is neither 1 nor −1, by the fundamental theorem of arithmeticFundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
, there exist primes pi, and positive integers ei, with
. In this case, the ideal (n) may be written as the intersection of the ideals (piei); that is, . This is referred to as a primary decomposition of the ideal (n).In general, an ideal Q of a ring is said to be primary
Primary ideal
In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n...
if Q is proper and whenever
, either or for some positive integer n. In , the primary ideals are precisely the ideals of the form (pe) where p is prime and e is a positive integer. Thus, a primary decomposition of (n) corresponds to representing (n) as the intersection of finitely many primary ideals.Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor −1 also asserts uniqueness of the representation
for pi prime and ei positive, a primary decomposition of (n) is essentially unique.For all of the above reasons, the following theorem, referred to as the Lasker–Noether theorem
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 written as an intersection of finitely many primary ideals...
, may be seen as a certain generalization of the fundamental theorem of arithmetic:
Theorem
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
Radical of an ideal
In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
; that is:
with Qi primary for all i and
for . Furthermore, if:
is decomposition of I with
for , and both decompositions of I are irredundant (meaning that no proper subset of either or yields an intersection equal to I), and (after possibly renumbering the Qi's) for all i.For any primary decomposition of I, the set of all radicals, that is, the set
remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module R/I; that is, the set of all annihilatorsAnnihilator (ring theory)
In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
of R/I (viewed as a module over R) that are prime.
Uses
The Noetherian property is central in ring theoryRing theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
and in areas that make heavy use of rings, such as algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. The reason behind this is that the Noetherian property is in some sense the ring-theoretic analogue of finiteness. For example, the fact that polynomial rings over a field are Noetherian allows one to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions.
Krull's principal ideal theorem
Krull's principal ideal theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull , gives a bound on the height of a principal ideal in a Noetherian ring...
is an important property of Noetherian rings. It states that every principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
in a commutative Noetherian ring has height one; that is, every principal ideal is contained in a prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
minimal amongst nonzero prime ideals. This early result was the first to suggest that Noetherian rings possessed a deep theory of dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
.
Examples
- Any field, including fields of rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, and complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. (A field only has two ideals — itself and (0).) - Any principal ideal domainPrincipal ideal domainIn abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
, such as the integers. - The ring of polynomials in finitely-many variables over the integers or a field.
Rings that are not Noetherian tend to be (in some sense) very large. Here are two examples of non-Noetherian rings:
- The ring of polynomials in infinitely-many variables, X1, X2, X3, etc. The sequence of ideals (X1), (X1, X2), (X1, X2, X3), etc. is ascending, and does not terminate.
- The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let In be the ideal of all continuous functions f such that f(x) = 0 for all x ≥ n. The sequence of ideals I0, I1, I2, etc., is an ascending chain that does not terminate.
However, a non-Noetherian ring can be a subring of a Noetherian ring:
- The ring of rational functions generated by x and y/xn over a field k is a subring of the field k(x,y) in only two variables.
Indeed, there are rings that are left Noetherian, but not right Noetherian, so that one must be careful in measuring the "size" of a ring this way.
Properties
- If R is a Noetherian ring, then R[X] is Noetherian by the Hilbert basis theorem. Also, R
X , the power series ring Formal power seriesIn mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
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.
- Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.)
- Every localizationLocalization of a ringIn abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
of a commutative Noetherian ring is Noetherian. - A consequence of the Akizuki-Hopkins-Levitzki TheoremHopkins–Levitzki theoremIn the branch of abstract algebra called ring theory, the Akizuki-Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R is called semiprimary if R/J is semisimple and J is a nilpotent ideal, where J denotes the...
is that every left Artinian ringArtinian ringIn abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
is left Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if right Artinian. The analogous statements with "right" and "left" interchanged are also true. - A ring R is left-Noetherian if and only if every finitely generated left R-moduleModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
is a Noetherian moduleNoetherian moduleIn abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....
. - A left Noetherian ring is left coherentCoherent ringIn mathematics, a coherent ring is a ring in which every finitely generated left ideal is finitely presented.Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings....
and a left Noetherian domainDomain (ring theory)In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
is a left Ore domain. - A ring is (left/right) Noetherian if and only if every direct sum of injective (left/right) modulesInjective moduleIn mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
is injective. Every injective module can be decomposed as direct sum of indecomposable injective modules.