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Abstract algebra module
In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially
Noetherian_module
Mathematical ring with well-behaved ideals
left-Noetherian if and only if every finitely generated left R-module is a Noetherian module. If a commutative ring admits a faithful Noetherian module over
Noetherian_ring
Index of articles associated with the same name
ring that satisfies the ascending chain condition on ideals. Noetherian module, a module that satisfies the ascending chain condition on submodules. More
Noetherian
In algebra, module with a finite generating set
generated module over a Noetherian ring is a Noetherian module (and indeed this property characterizes Noetherian rings): A module over a Noetherian ring is
Finitely_generated_module
Module which satisfies the descending chain condition on submodules
Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian
Artinian_module
In algebra, integer associated to a module
algebraic geometry, a module over a Noetherian commutative ring R can have finite length only when the module has Krull dimension zero. Modules of finite length
Length_of_a_module
Generalization of vector spaces from fields to rings
the ring, equivalently rm = 0 implies r = 0 or m = 0. Noetherian A Noetherian module is a module that satisfies the ascending chain condition on submodules
Module_(mathematics)
Mathematical object in abstract algebra
modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian
Injective_module
Theorem in abstract algebra
generated as a module over A {\displaystyle A} , if B {\displaystyle B} is a Noetherian ring, then A {\displaystyle A} is a Noetherian ring. (Note the
Eakin–Nagata_theorem
Ring in abstract algebra
left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module. An integral domain
Artinian_ring
In algebra, expression of an ideal as the intersection of ideals of a specific type
a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of
Primary_decomposition
Theorem in algebra
algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring
Matlis_duality
Statement in abstract algebra
indecomposable modules, and thus every finitely generated module over a PID is a completely decomposable module. Since PID's are Noetherian rings, this can
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Direct summand of a free module (mathematics)
is true for finitely generated modules over Noetherian rings: a finitely generated module over a commutative Noetherian ring is locally free if and only
Projective_module
German mathematician (1882–1935)
A Noetherian module is a module in which every strictly ascending chain of submodules becomes constant after a finite number of steps. A Noetherian space
Emmy_Noether
Prime ideal that is an annihilator of a prime submodule
of R. For a Noetherian module M over any ring, there are only finitely many associated primes of M. For the case for commutative Noetherian rings, see
Associated_prime
Skolem–Noether theorem Noetherian Noetherian group Noetherian induction Noetherian module Noetherian ring Noetherian scheme Noetherian topological space "Noether
List of things named after Emmy Noether
List_of_things_named_after_Emmy_Noether
Algebraic structure in ring theory
is projective. Over a Noetherian ring, every finitely generated flat module is projective, since every finitely generated module is finitely presented
Flat_module
conjecture holds in Noetherian serial rings. Any simple module is trivially uniserial, and likewise semisimple modules are serial modules. Many examples of
Serial_module
orders in a semisimple ring. Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules. In the literature, uniform dimension
Uniform_module
Direct sum of irreducible modules
area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from
Semisimple_module
In mathematics, dimension of a ring
dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings
Krull_dimension
In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It
Artin–Rees_lemma
torsion-free module M is an ideal isomorphic (as a module) to a torsion-free quotient of M by a free submodule. Buchsbaum ring A Buchsbaum ring is a Noetherian local
Glossary of commutative algebra
Glossary_of_commutative_algebra
Theorem in algebra mathematics
arbitrary modules U over R, for U need not contain any maximal submodules. Naturally, if U is a Noetherian module, this holds. If R is Noetherian, and U
Nakayama's_lemma
states that if R is a semiprimary ring and M is an R-module, the three module conditions Noetherian, Artinian and "has a composition series" are equivalent
Hopkins–Levitzki_theorem
Type of commutative ring in mathematics
commutative Noetherian local ring R, a finite (i.e. finitely generated) R-module M ≠ 0 {\displaystyle M\neq 0} is a Cohen–Macaulay module if d e p t h
Cohen–Macaulay_ring
finite module.[citation needed] This prompted Zariski (1948) to ask whether a local Noetherian domain such that its integral closure is a finite module is
Analytically_unramified_ring
isomorphic to R as a module, so its support is the entire space: Supp(I) = Spec(R). The support of a finite module over a Noetherian ring is always closed
Support_of_a_module
Algebraic structure
arbitrary modules U over R, for U need not contain any maximal submodules. Naturally, if U is a Noetherian module, this holds. If R is Noetherian, and U
Noncommutative_ring
Module over a ring
Over a Noetherian integral domain, torsion-free modules are the modules whose only associated prime is zero. More generally, over a Noetherian commutative
Torsion-free_module
Tate and his former advisor Emil Artin, states: Let A be a commutative Noetherian ring and B ⊂ C {\displaystyle B\subset C} commutative algebras over A
Artin–Tate_lemma
Invariant of rings and modules
and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local
Depth_(ring_theory)
endomorphism is an endomorphism, some power of which is zero. Noetherian A Noetherian module is a module such that every submodule is finitely generated. Equivalently
Glossary_of_module_theory
pseudo-geometric ring if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a
Nagata_ring
Branch of algebra that studies commutative rings
modern approach to commutative algebra using module theory is usually credited to Krull and Noether. A Noetherian ring, named after Emmy Noether, is a ring
Commutative_algebra
Local ring in commutative algebra
Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some
Gorenstein_ring
Decomposition of an algebraic structure
an Artinian module and a Noetherian module. If R is an Artinian ring, then every finitely generated R-module is Artinian and Noetherian, and thus has
Composition_series
Sheaf consisting of modules on a ringed space; generalizing vector bundles
piece) and M a graded R-module. Let X be the Proj of R (so X is a projective scheme if R is Noetherian). Then there is an O-module M~ such that for any homogeneous
Sheaf_of_modules
to the fact that rad(M) is the sum of superfluous submodules, in a Noetherian module, rad(M) itself is a superfluous submodule. In fact, if M is finitely
Radical_of_a_module
Algebraic structure
In particular, Noetherian rings (see also § Noetherian rings, below) can be defined as the rings such that every submodule of a module of finite type
Commutative_ring
Construction of a ring of fractions
"denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R so that it consists of fractions
Localization (commutative algebra)
Localization_(commutative_algebra)
Algebraic formula
3.7), states that if R is a commutative Noetherian local ring and M is a non-zero finitely generated R-module of finite projective dimension, then: p
Auslander–Buchsbaum_formula
in Grothendieck local duality. A dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m, the R/m
Dualizing_module
Ideal that maps to zero a subset of a module
a left R-module. When R is commutative and Noetherian, the set DR is precisely equal to the union of the associated primes of the R-module R. Faltings'
Annihilator_(ring_theory)
Concept in ring theory and homological algebra
commutative Noetherian local rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose definition is module-theoretic
Global_dimension
Algebraic structure with addition and multiplication
that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian. For commutative
Ring_(mathematics)
Zero divisors in a module
submodule of M for all right R-modules. Since right Noetherian domains are Ore, this covers the case when R is a right Noetherian domain (which might not be
Torsion_(algebra)
(Mathematical) ring with a unique maximal ideal
authors required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article
Local_ring
A-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if A is Noetherian, a
Perfect_complex
minimal number of generators of a finitely generated module M {\displaystyle M} over a commutative Noetherian ring. The usefulness of the theorem stems from
Forster–Swan_theorem
In algebra, completion w.r.t. powers of an ideal
are modules over R^. The completion of a Noetherian ring R is a flat module over R. The completion of a finitely generated module M over a Noetherian ring
Completion_of_a_ring
that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal- and module-theoretic properties of Dedekind domains,
Prüfer_domain
Mathematical theorem
indecomposable projective modules over semiperfect rings. In general, the theorem fails if one only assumes that the module is Noetherian or Artinian. The present-day
Krull–Schmidt_theorem
Mathematical object
being hopfian or cohopfian as a ring. A Noetherian module is hopfian, and an Artinian module is cohopfian. The module RR is hopfian if and only if R is a
Hopfian_object
Type of mathematical equation
syzygy module depends on the chosen generating set, most of its properties are independent; see § Stable properties, below. If the ring R is Noetherian, or
Linear_relation
Module over a sheaf of differential operators
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of
D-module
On the effects of changing the ring of ''K''-groups
regular noetherian ring R {\displaystyle R} , every finitely generated module admits a finite resolution by finitely generated projective modules: 0 ⟶ P
Fundamental theorem of algebraic K-theory
Fundamental_theorem_of_algebraic_K-theory
Notion in abstract algebra
4153/CJM-1963-041-4, ISSN 0008-414X, MR 0147509 Matlis, Eben (1958), "Injective modules over Noetherian rings", Pacific Journal of Mathematics, 8 (3): 511–528, doi:10
Injective_hull
Concept in commutative algebra
topology is called separated. By Krull's intersection theorem, if R is a Noetherian ring which is an integral domain or a local ring, it holds that ⋂n>0𝔞n=0
I-adic_topology
Branch of mathematics that studies algebraic structures
theory) Simple module, Semisimple module Indecomposable module Artinian module, Noetherian module Homological types: Projective module Projective cover
List of abstract algebra topics
List_of_abstract_algebra_topics
Type of Abelian category (in category theory in mathematics)
the familiar notion of Noetherian modules.) A Grothendieck category is called locally Noetherian if it has a set of Noetherian generators; an example
Grothendieck_category
Generalization of vector bundles
the module M = Γ ( U , F ) {\displaystyle M=\Gamma (U,{\mathcal {F}})} over A {\displaystyle A} . When X {\displaystyle X} is a locally Noetherian scheme
Coherent_sheaf
In abstract algebra, the weak dimension of a nonzero right module M over a ring R is the largest number n such that the Tor group Tor n R ( M , N ) {\displaystyle
Weak_dimension
Study of dimension in algebraic geometry
E.D. Let R {\displaystyle R} be a noetherian ring. The projective dimension of a finite R {\displaystyle R} -module M {\displaystyle M} is the shortest
Dimension_theory_(algebra)
Branch of algebra
conditions for a Noetherian ring to be an Artinian ring Morita theory consists of theorems determining when two rings have "equivalent" module categories Cartan–Brauer–Hua
Ring_theory
Algebraic structure
discrete valuation ring. A noetherian ring is a Krull domain if and only if it is an integrally closed domain. In the non-noetherian setting, one has the following:
Integrally_closed_domain
R be a Noetherian ring and M a reflexive finitely generated module over R. Then M ⊗ R S {\displaystyle M\otimes _{R}S} is a reflexive module over S whenever
Torsionless_module
Well-behaved sequence in a commutative ring
maximal ideal. For a Noetherian local ring R, the depth of the zero module is ∞, whereas the depth of a nonzero finitely generated R-module M is at most the
Regular_sequence
Algebraic structure
finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings. Every left Noetherian ring is left coherent
Coherent_ring
Construction in homological algebra
})\end{aligned}}} Let A {\displaystyle A} be a finitely generated module over a commutative Noetherian ring R {\displaystyle R} . Then Ext {\displaystyle \operatorname
Ext_functor
Integral domain in which the sum of two principal ideals is again a principal ideal
ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals; if so, it is not
Bézout_domain
Abelian group in which every element can, in some sense, be divided by positive integers
like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because
Divisible_group
Equivalence relation on rings
their modules, as modules can be viewed as representations of rings. Every ring R has a natural R-module structure on itself where the module action
Morita_equivalence
Generalization of algebraic variety
a coherent sheaf (on a Noetherian scheme X, say) is an OX-module that is the sheaf associated to a finitely generated module on each affine open subset
Scheme_(mathematics)
Construction in commutative algebra
between R and its associated graded ring grIR. Assume R is Noetherian; then R[It] is also Noetherian. The Krull dimension of the Rees algebra is dim R [
Rees_algebra
spaces. Each module in category O {\displaystyle {\mathcal {O}}} is a Noetherian module. O {\displaystyle {\mathcal {O}}} is an abelian category. O {\displaystyle
Category_O
Hilbert and Pierre Samuel, of a nonzero finitely generated module M over a commutative Noetherian local ring A and a primary ideal I of A is the map χMI:ℕ→ℕ
Hilbert–Samuel_function
Type of ring in commutative algebra
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal
Regular_local_ring
}I^{n}/I^{n+1}} . Similarly, if M is a left R-module, then the associated graded module is the graded module over gr I R {\displaystyle \operatorname {gr}
Associated_graded_ring
Mathematical technique in algebraic geometry
French for unscrewing. Let X be a noetherian scheme. Let C be a collection of objects of the category of coherent OX-modules which contains the zero sheaf
Dévissage
Mathematical concept in dimension theory of local rings
In mathematics, a system of parameters for a local Noetherian ring of Krull dimension d with maximal ideal m is a set of elements x1, ..., xd that satisfies
System_of_parameters
Algebraic structure
can be written in the form ax + by, etc. Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and
Principal_ideal_domain
In mathematics, element that equals its square
can be achieved in many ways, such as requiring the ring to be right Noetherian. If a decomposition R = c1R ⊕ c2R ⊕ ... ⊕ cnR exists with each ci a centrally
Idempotent_(ring_theory)
On polynomial rings over fields
between the generators of an ideal, or, more generally, a module. As the relations form a module, one may consider the relations between the relations; the
Hilbert's_syzygy_theorem
Mathematical element
subrings that are finitely generated A {\displaystyle A} -modules. If A {\displaystyle A} is noetherian, transitivity of integrality can be weakened to the
Integral_element
Algebra with unique prime factorization
is the fact that being a Dedekind domain is, among Noetherian domains, a local property: a Noetherian domain R {\displaystyle R} is Dedekind iff for every
Dedekind_domain
Would relate vector bundles over a regular Noetherian ring and over a polynomial ring
mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring A [ t 1 , … , t n ] {\displaystyle
Bass–Quillen_conjecture
group. There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a dualizing complex. Matlis
Grothendieck_local_duality
of modules on a scheme is flat or free. They are due to Alexander Grothendieck. Generic flatness states that if Y is an integral locally noetherian scheme
Generic_flatness
2-dimensional local domains are catenary. The ring A is Noetherian because B is Noetherian and is a finite A-module. However A is not universally catenary, because
Catenary_ring
Mathematical term in group theory
Z-module), ℤ(p∞) is Artinian but not Noetherian. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas
Prüfer_group
Scheme theory concept
noetherian, then Y is locally integral. If f is faithfully flat and quasi-compact, and if X is locally noetherian, then Y is also locally noetherian.
Flat_morphism
Vector bundle of rank 1
finitely generated projective module M over a Noetherian domain and the resulting invertible module is called the determinant module of M. The first Stiefel–Whitney
Line_bundle
Type of algebra
commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, A is a Noetherian ring. Finitely generated reduced
Finitely_generated_algebra
In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence. A sequence ( a 1 , ⋯ , a r ) {\displaystyle
Buchsbaum_ring
IBN, as does any left-Noetherian ring and any semilocal ring. Proof Let A be a commutative ring and assume there exists an A-module isomorphism f : A n
Invariant_basis_number
Endomorphism algebra of an abelian group
ring of a nonzero right uniserial module has either one or two maximal right ideals. If the module is Artinian, Noetherian, projective or injective, then
Endomorphism_ring
Commutative algebra theorem
Serge Lang's Algebra. A generalization relating projective modules over regular Noetherian rings A and their polynomial rings is known as the Bass–Quillen
Quillen–Suslin_theorem
Result in ring theory
particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all
Goldie's_theorem
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NOETHERIAN MODULE
NOETHERIAN MODULE
NOETHERIAN MODULE
NOETHERIAN MODULE
NOETHERIAN MODULE
NOETHERIAN MODULE
NOETHERIAN MODULE
NOETHERIAN MODULE
NOETHERIAN MODULE
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