Right-Noetherian Ring
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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 I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. 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 a_1, \ldots , a_n in ''I'' such that I=Ra_1 + \cdots + Ra_n.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 f_1, f_2, \dots of elements in ''R'', there exists an integer n such that each f_i is a finite linear combination f_i = \sum_^n r_j f_j with coefficients r_j 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 ...
R /math> is Noetherian by the Hilbert's basis theorem. By induction, R _1, \ldots, X_n/math> is a Noetherian ring. Also, , the
power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
, is a Noetherian ring. * If is a Noetherian ring and is a two-sided ideal, then the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
is also Noetherian. Stated differently, the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of any
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition prese ...
of a Noetherian ring is Noetherian. * Every finitely-generated
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.) * A ring ''R'' is left-Noetherian
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
every finitely generated left ''R''-module is a
Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the pr ...
. * If a commutative ring admits a faithful Noetherian module over it, then the ring is a Noetherian ring. * ( Eakin–Nagata) If a ring ''A'' is a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of a commutative Noetherian ring ''B'' such that ''B'' is a finitely generated module over ''A'', then ''A'' is a Noetherian ring. *Similarly, if a ring ''A'' is a subring of a commutative Noetherian ring ''B'' such that ''B'' is faithfully flat over ''A'' (or more generally exhibits ''A'' as a pure subring), then ''A'' is a Noetherian ring (see the "faithfully flat" article for the reasoning). * Every localization of a commutative Noetherian ring is Noetherian. * A consequence of the Akizuki–Hopkins–Levitzki theorem is that every left
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...
is left Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if it is right Artinian. The analogous statements with "right" and "left" interchanged are also true. * A left Noetherian ring is left coherent and a left Noetherian
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...
is a left Ore domain. * (Bass) A ring is (left/right) Noetherian if and only if every
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
(left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of
indecomposable Indecomposability or indecomposable may refer to any of several subjects in mathematics: * Indecomposable module, in algebra * Indecomposable distribution, in probability * Indecomposable continuum, in topology * Indecomposability (intuitionist ...
injective modules. See also #Implication on injective modules below. * In a commutative Noetherian ring, there are only finitely many
minimal prime ideal In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Defini ...
s. Also, the
descending 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 ...
holds on prime ideals. * In a commutative Noetherian domain ''R'', every element can be factorized into
irreducible element In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements. Relationship with prime elements Irreducible elements should not be confus ...
s (in short, ''R'' is a factorization domain). Thus, if, in addition, the factorization is unique
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
multiplication of the factors by
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
s, then ''R'' is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
.


Examples

* Any field, including the fields of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s,
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, and
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, is Noetherian. (A field only has two ideals — itself and (0).) * Any
principal ideal ring In mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called pri ...
, such as the integers, is Noetherian since every ideal is generated by a single element. This includes
principal ideal domain In mathematics, 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 princip ...
s and
Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. ...
s. * A
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
(e.g., rings of integers) is a Noetherian domain in which every ideal is generated by at most two elements. * The
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ide ...
of an
affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ide ...
is a Noetherian ring, as a consequence of the Hilbert basis theorem. * The enveloping algebra ''U'' of a finite-dimensional
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
\mathfrak is a both left and right Noetherian ring; this follows from the fact that the associated graded ring of ''U'' is a quotient of \operatorname(\mathfrak), which is a polynomial ring over a field; thus, Noetherian. For the same reason, the
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More pre ...
, and more general rings of differential operators, are Noetherian. * The ring of polynomials in finitely-many variables over the integers or a field is Noetherian. Rings that are not Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings: * The ring of polynomials in infinitely-many variables, ''X''1, ''X''2, ''X''3, etc. The sequence of ideals (''X''1), (''X''1, ''X''2), (''X''1, ''X''2, ''X''3), etc. is ascending, and does not terminate. * The ring of all
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficien ...
s is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (21/2), (21/4), (21/8), ... * 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 ''I''0, ''I''1, ''I''2, etc., is an ascending chain that does not terminate. * The ring of
stable homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure ...
is not Noetherian. However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
is a subring of a field, any integral domain that is not Noetherian provides an example. To give a less trivial example, * The ring of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s generated by ''x'' and ''y'' /''x''''n'' over a field ''k'' is a subring of the field ''k''(''x'',''y'') in only two variables. Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. For example, if ''L'' is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of Q2 isomorphic to Z, let ''R'' be the ring of homomorphisms ''f'' from Q2 to itself satisfying ''f''(''L'') ⊂ ''L''. Choosing a basis, we can describe the same ring ''R'' as :R=\left\. This ring is right Noetherian, but not left Noetherian; the subset ''I'' ⊂ ''R'' consisting of elements with ''a'' = 0 and ''γ'' = 0 is a left ideal that is not finitely generated as a left ''R''-module. If ''R'' is a commutative subring of a left Noetherian ring ''S'', and ''S'' is finitely generated as a left ''R''-module, then ''R'' is Noetherian. (In the special case when ''S'' is commutative, this is known as Eakin's theorem.) However, this is not true if ''R'' is not commutative: the ring ''R'' of the previous paragraph is a subring of the left Noetherian ring ''S'' = Hom(Q2, Q2), and ''S'' is finitely generated as a left ''R''-module, but ''R'' is not left Noetherian. A
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
is not necessarily a Noetherian ring. It does satisfy a weaker condition: the
ascending chain condition on principal ideals In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals (abbreviat ...
. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain. A
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' such ...
is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian.


Key theorems

Many important theorems in ring theory (especially the theory of commutative rings) rely on the assumptions that the rings are Noetherian.


Commutative case

*Over a commutative Noetherian ring, each ideal has a
primary decomposition 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 rela ...
, meaning that it can be written as an
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of finitely many
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 ''y'n'' is also an element of ''Q'', for some ''n'' > 0. ...
s (whose
radical Radical may refer to: Politics and ideology Politics * Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...
s are all distinct) where an ideal ''Q'' is called primary if it is
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y'' ''n'' ∈ ''Q'' for some positive integer ''n''. For example, if an element f = p_1^ \cdots p_r^ is a product of powers of distinct prime elements, then (f) = (p_1^) \cap \cdots \cap (p_r^) and thus the primary decomposition is a direct generalization of
prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...
of integers and polynomials. *A Noetherian ring is defined in terms of ascending chains of ideals. The Artin–Rees lemma, on the other hand, gives some information about a descending chain of ideals given by powers of ideals I \supseteq I^2 \supseteq I^3 \supseteq \cdots . It is a technical tool that is used to prove other key theorems such as 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 ...
. *The
dimension theory In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem,
Krull's principal ideal theorem 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 commutative Noetherian ring. The theorem is sometimes referred to by its German name, '' ...
, already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and (Noetherian) universally catenary rings, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary.


Non-commutative case

*
Goldie's theorem In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring ''R'' that has finite uniform dimension (="finite rank") as a right module ...


Implication on injective modules

Given a ring, there is a close connection between the behaviors of
injective module In 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. Specifically, if ''Q'' is a submodule ...
s over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring ''R'', the following are equivalent: *''R'' is a left Noetherian ring. *(Bass) Each direct sum of injective left ''R''-modules is injective. *Each injective left ''R''-module is a direct sum of
indecomposable Indecomposability or indecomposable may refer to any of several subjects in mathematics: * Indecomposable module, in algebra * Indecomposable distribution, in probability * Indecomposable continuum, in topology * Indecomposability (intuitionist ...
injective modules. *(Faith–Walker) There exists a
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
\mathfrak such that each injective left module over ''R'' is a direct sum of \mathfrak-generated modules (a module is \mathfrak-generated if it has a
generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied t ...
of
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
at most \mathfrak). *There exists a left ''R''-module ''H'' such that every left ''R''-module embeds into a direct sum of copies of ''H''. The
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of an indecomposable injective module is
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...
and thus Azumaya's theorem says that, over a left Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the
Krull–Schmidt theorem In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chain A chain is a wikt:series#Noun, serial assembly of connected pieces, called links, typically made of metal, with an overall ...
).


See also

*
Noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. T ...
*
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...


Notes


References

* * Atiyah, M. F., MacDonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley-Longman. * Nicolas Bourbaki, Commutative algebra * * * * * Chapter X of *


External links

* {{springer, title=Noetherian ring, id=p/n066850 Ring theory