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In algebra, module with a finite generating set
a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module
Finitely_generated_module
Direct summand of a free module (mathematics)
a finitely generated projective A-module containing A as a subring, then A is a direct factor of B. Let P be a finitely generated projective module over
Projective_module
Statement in abstract algebra
theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms. For every finitely generated module M over
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
In mathematics, a module that has a basis
infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group). A finitely generated module over a commutative local ring is
Free_module
Algebraic structure in ring theory
f:F\to M,} where F {\displaystyle F} is a finitely generated free R-module, and for every finitely generated R-submodule K {\displaystyle K} of ker f
Flat_module
Topological structure in number theory
sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module is a torsion module, which happens
Iwasawa_algebra
Type of algebra
mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R {\displaystyle
Finitely_generated_algebra
Group type in algebra
cardinality of a generating set for the group. By definition, the rank of a finitely generated group is finite. Finitely generated module Presentation of
Finitely_generated_group
In algebra, expression of an ideal as the intersection of ideals of a specific type
a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over
Primary_decomposition
Theorem in algebra mathematics
ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative
Nakayama's_lemma
Abstract algebra module
with finitely generated modules: a submodule of a finitely generated module need not be finitely generated. The integers, considered as a module over
Noetherian_module
Algebra with unique prime factorization
particular, P {\displaystyle P} is a finitely generated free module. Now let M {\displaystyle M} be a finitely generated module over an arbitrary Dedekind domain
Dedekind_domain
Generalization of vector spaces from fields to rings
denoted by R-Mod (see category of modules). Finitely generated An R-module M is finitely generated if there exist finitely many elements x1, ..., xn in M
Module_(mathematics)
Submodule of a mathematical ring
nonzero finitely generated module admits a maximal submodule, in particular, one has: If J M = M {\displaystyle JM=M} and M is finitely generated, then
Ideal_(ring_theory)
In algebra, integer associated to a module
have finite length only when the module has Krull dimension zero. Modules of finite length are finitely generated modules, but most finitely generated modules
Length_of_a_module
Commutative group (mathematics)
classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal
Abelian_group
Commutative group where every element is the sum of elements from one finite subset
a finite (hence finitely generated) abelian group. Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian
Finitely generated abelian group
Finitely_generated_abelian_group
Concept in mathematics
over itself. If there is a finite generating set, then a module is said to be finitely generated. This applies to ideals, which are the submodules of the
Generating_set_of_a_module
Algebraic structure with only one element
κ ∈ R. The most general of them, the zero module, is a finitely-generated module with an empty generating set. For structures requiring the multiplication
Zero_object_(algebra)
On polynomial rings over fields
free resolution. Hilbert's syzygy theorem states that, if M is a finitely generated module over a polynomial ring k [ x 1 , … , x n ] {\displaystyle k[x_{1}
Hilbert's_syzygy_theorem
are finitely generated free modules. 2. A finitely presented module is a module that admits a finite free presentation. finitely generated A module M {\displaystyle
Glossary_of_module_theory
Concept in category theory
group. Finitely generated group Finitely generated monoid Finitely generated abelian group Finitely generated module Finitely generated ideal Finitely generated
Finitely_generated_object
Concept in algebraic geometry
makes Ai a finitely generated module over Bi (in other words, a finite Bi-algebra). One also says that X is finite over Y. In fact, f is finite if and only
Finite_morphism
Module over a ring
finitely-generated free module and an ideal, and the class of the ideal is uniquely determined by the module. Over a principal ideal domain, finitely-generated
Torsion-free_module
more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization
Cyclic_module
much weaker than saying it is finitely generated as a module. 3. An extension of fields is called finitely generated if elements of the larger field
Glossary of commutative algebra
Glossary_of_commutative_algebra
Representation of an algebra as a free module
decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions
Hironaka_decomposition
Type of mathematical equation
if M is finitely generated, then the syzygy module is also finitely generated. A syzygy module of this syzygy module is a second syzygy module of M. Continuing
Linear_relation
Ideal that maps to zero a subset of a module
Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator
Annihilator_(ring_theory)
Algebraic structure
any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian
Commutative_ring
be coherent, since finitely generated projective modules are finitely related. Though ideals of Dedekind domains can all be generated by two elements, for
Prüfer_domain
over a ring R {\displaystyle R} is called finite if it is finitely generated as an R {\displaystyle R} -module. An R {\displaystyle R} -algebra can be thought
Finite_algebra
Module which satisfies the descending chain condition on submodules
and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both
Artinian_module
Zero divisors in a module
principal ideal domain and M is a finitely generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives
Torsion_(algebra)
the Fitting ideals of a finitely generated module over a commutative ring describe the obstructions to generating the module by a given number of elements
Fitting_ideal
Branch of mathematics
basis. The modules that have a basis are the free modules, and those that are spanned by a finite set are the finitely generated modules. Module homomorphisms
Linear_algebra
type" means "finitely generated algebra" and "finite" means "finitely generated module".) The lemma was introduced by E. Artin and J. Tate in 1951 to
Artin–Tate_lemma
Result of commutative algebra
over k {\displaystyle k} and such that A {\displaystyle A} is a finitely generated module over the polynomial ring S = k [ y 1 , y 2 , … , y d ] {\displaystyle
Noether_normalization_lemma
quotient field is a finitely generated A {\displaystyle A} -module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L {\displaystyle
Nagata_ring
In algebra, a module over a ring
of the standard basis generates M. In particular, if J is finite, then M is a finitely generated module. If I and J are finite sets, then the presentation
Free_presentation
sheaves. Let I be an ideal in a Noetherian ring R; let M be a finitely generated R-module and let N a submodule of M. Then there exists an integer k ≥ 1
Artin–Rees_lemma
Matrix normal form
structure theorem for finitely generated modules over a principal ideal domain, which includes the fundamental theorem of finitely generated abelian groups.
Smith_normal_form
Mathematical ring with well-behaved ideals
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
Noetherian_ring
algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin. Every Artin algebra is an Artin
Artin_algebra
infinite. Finitely-generated modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules over a principal
Indecomposable_module
Mathematical object in abstract algebra
k with finite dimension over k, then Homk(−, k) is a duality between finitely generated left A-modules and finitely generated right A-modules. Therefore
Injective_module
Topics referred to by the same term
finitely presented may refer to: finitely presented group finitely presented monoid finitely presented module finitely presented algebra finitely presented
Finitely_presented
Prime ideal that is an annihilator of a prime submodule
primes. A module is called coprimary if xm = 0 for some nonzero m ∈ M implies xnM = 0 for some positive integer n. A nonzero finitely generated module M over
Associated_prime
invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal
Invariant_factor
Module over the non-commutative Dieudonné ring
Dieudonné modules over an algebraically closed field k {\displaystyle k} up to "isogeny". More precisely, it classifies the finitely generated modules over
Dieudonné_module
Linear map over a ring
→ M {\displaystyle \phi :M\to M} be an endomorphism between finitely generated R-modules for a commutative ring R. Then ϕ {\displaystyle \phi } is killed
Module_homomorphism
a finitely generated projective module and hence is projective (semi-hereditary condition). Conversely, over a Dedekind domain, a finitely generated torsion-free
Torsionless_module
Algebraic structure
for finitely generated modules over a principal ideal domain applies to K[X], when K is a field. This means that every finitely generated module over
Polynomial_ring
states an upper bound for the minimal number of generators of a finitely generated module M {\displaystyle M} over a commutative Noetherian ring. The usefulness
Forster–Swan_theorem
Ideal of a ring contained in no other ideal except the ring itself
with unity, a nonzero module does not necessarily have maximal submodules. However, as noted above, finitely generated nonzero modules have maximal submodules
Maximal_ideal
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
Ring in abstract algebra
a left module over a left Artinian ring. Then the following are equivalent (Hopkins' theorem): (i) M is finitely generated, (ii) M has finite length (i
Artinian_ring
M} is a finitely generated module if and only if the cosocle M / r a d ( M ) {\displaystyle M/\mathrm {rad} (M)} is finitely generated and r a d ( M )
Radical_of_a_module
Construction in homological algebra
}\operatorname {Ext} _{R}^{i}(M,N_{\alpha })\end{aligned}}} Let A be a finitely generated module over a commutative Noetherian ring R. Then Ext commutes with localization
Ext_functor
Branch of mathematics that studies algebraic structures
Flat module Flat cover Coherent module Finitely-generated module Finitely-presented module Finitely related module Algebraically compact module Reflexive
List of abstract algebra topics
List_of_abstract_algebra_topics
free module is a module which is close to being free. A module M over a ring R is stably free if there exists a free finitely generated module F over
Stably_free_module
function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module M {\displaystyle M} over a commutative Noetherian local ring A
Hilbert–Samuel_function
Algebraic structure with addition and multiplication
concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated
Ring_(mathematics)
algebraic geometry. Formally, a finitely generated module M over a ring R is said to be invertible if it is locally a free module of rank 1. In other words
Invertible_module
Subject area in mathematics
Indeed, the global dimension of regular rings is finite, i.e. any finitely generated module has a finite projective resolution P* → M, and a simple argument
Algebraic_K-theory
{\displaystyle A\subset B} such that B {\displaystyle B} is finitely generated as a module over A {\displaystyle A} , if B {\displaystyle B} is a Noetherian
Eakin–Nagata_theorem
Construction in homological algebra
xn elements of R and I = (x1, ..., xn) the ideal generated by them. For a finitely generated module M over R, if, for some integer m, H i ( K ( x 1
Koszul_complex
In mathematics, dimension of a ring
schemes, finitely generated modules are interpreted as coherent sheaves, or generalized finite rank vector bundles. The Krull dimension of a module over a
Krull_dimension
Theorem in algebra
then the Matlis module is K/R. In the special case when R is the ring of p-adic numbers, the Matlis dual of a finitely-generated module is the Pontryagin
Matlis_duality
Algebraic formula
is a PID and M {\displaystyle M} a finitely generated R {\displaystyle R} -module, then M is isomorphic to a finite direct sum of the form M ≅ R r ⊕ ⨁
Elementary_divisors
Mathematical element
generated by A and b is a finitely generated A-module; (iii) there exists a subring C of B containing A[b] and which is a finitely generated A-module;
Integral_element
every finitely-generated R-algebra S lying between R and the field of fractions K of R, the integral closure of S in K is a finitely generated module over
Analytically_unramified_ring
Topics referred to by the same term
refers to writing a finitely generated module over a principal ideal domain as the direct sum of cyclic modules and one free module. This disambiguation
Cycle_decomposition
Algebraic formula
a commutative Noetherian local ring and M is a non-zero finitely generated R-module of finite projective dimension, then: p d R ( M ) + d e p t h ( M
Auslander–Buchsbaum_formula
Analysis of datasets using techniques from topology
language of commutative algebra appeared in 2005: for a finitely generated persistence module C {\displaystyle C} with field F {\displaystyle F} coefficients
Topological_data_analysis
Ring without non-zero nilpotent elements
all minimal prime ideals. Over a Noetherian ring R, we say a finitely generated module M has locally constant rank if p ↦ dim k ( p ) ( M ⊗ k ( p )
Reduced_ring
Scalar-valued bilinear function
notion for a module over a commutative ring is that a bilinear form is unimodular if V → V∗ is an isomorphism. Given a finitely generated module over a commutative
Bilinear_form
Integral domain in which the sum of two principal ideals is again a principal ideal
Bézout's identity holds for every pair of elements, and that every finitely generated ideal is principal. Bézout domains are a form of Prüfer domain. Any
Bézout_domain
persistence modules." The case when P {\displaystyle P} is finite is a straightforward application of the structure theorem for finitely generated modules over
Persistence_module
vector space over K (and thus also an R-module). An R-submodule M of a V is called a lattice if M is finitely generated over R. It is called full if V = K
Lattice_(module)
Sheaf consisting of modules on a ringed space; generalizing vector bundles
n, the Serre twist F(n) is generated by finitely many global sections. Moreover, For each i, Hi(X, F) is finitely generated over R0, and There is an integer
Sheaf_of_modules
algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S:=k[y1, y2, ..., yd]. The theorem has
List of inventions and discoveries by women
List_of_inventions_and_discoveries_by_women
Mathematical property
infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely presented groups. Given an integer n ≥ 1, a group
Finiteness properties of groups
Finiteness_properties_of_groups
Central object of study in category theory
preserve the direct sum decomposition – see Structure theorem for finitely generated modules over a principal ideal domain § Uniqueness for example. Some authors
Natural_transformation
Abelian group extending a commutative monoid
sequences of finitely generated modules) and K 0 ( R ) {\displaystyle K_{0}(R)} (defined via direct sum of finitely generated projective modules) coincide. In
Grothendieck_group
Algebraic structure
finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely
Coherent_ring
finitely generated module have finite hollow dimension? The answer turns out to be no: it was shown in (Sarath & Varadarajan 1979) that if a module M
Uniform_module
Index of articles associated with the same name
{\displaystyle R} is a finite-dimensional unital algebra and M {\displaystyle M} a finitely generated R {\displaystyle R} -module then the socle consists
Socle_(mathematics)
resolution of a finitely-generated module M over a Noetherian ring: for each prime ideal p there is a corresponding indecomposable injective module, and the
Bass_number
Module over a sheaf of differential operators
ring in 2n indeterminates. In particular it is commutative. Finitely generated D-modules M are endowed with so-called "good" filtrations F∗M, which are
D-module
a local ring (#Characterization of a local ring). For a finitely generated projective module over a commutative local ring, the theorem is an easy consequence
Kaplansky's theorem on projective modules
Kaplansky's_theorem_on_projective_modules
Generalization of algebraic variety
(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 of X. Coherent
Scheme_(mathematics)
Number of elements in a subset of a commutative group
defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. For this, see finitely generated module#Generic
Rank_of_an_abelian_group
Concept in algebra
{\displaystyle I,J} are comaximal. Let M {\displaystyle M} be a finitely generated module over a Noetherian ring R {\displaystyle R} . Then ann R ( M
Radical_of_an_ideal
Mathematical group based upon a finite number of elements
Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of
Finite_group
For example, if A is Noetherian, a module over A is perfect if and only if it is finitely generated and of finite projective dimension. Perfect complexes
Perfect_complex
simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions
Compact_element
In algebra, completion w.r.t. powers of an ideal
converts any module over R into a complete topological module over R ^ I {\displaystyle {\widehat {R}}_{I}} if I is finitely generated. The ring of p-adic
Completion_of_a_ring
Category with direct sums and certain types of kernels and cokernels
category of modules (Mitchell's embedding theorem). If R is a left-noetherian ring, then the category of finitely generated left modules over R is abelian
Abelian_category
(right) R-module has a projective cover. Every finitely generated left (right) R-module has a projective cover. The category of finitely generated projective
Perfect_ring
FINITELY GENERATED-MODULE
FINITELY GENERATED-MODULE
Boy/Male
Indian, Sanskrit
Devoted; Venerated
Girl/Female
Biblical
Penetrated.
Boy/Male
American, Australian, Christian, Danish, French, Gaelic, Latin, Scottish
Son of; Taken from Mackenzie; Greatest; Finely Made; Comely
Boy/Male
Arabic, Muslim, Sindhi
Venerated; Honoured
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Sindhi, Tamil, Telugu
Virtuous; Venerated
Boy/Male
Muslim
Honored, Venerated
Girl/Female
Tamil
Generates harmony in dance and music
Biblical
penetrated
Girl/Female
African, American, British, Celtic, Chinese, Christian, English, Irish
Handsome; Beautiful; Knowing; Good Looking; Born of Fire; Finely Made
Girl/Female
Tamil
Aninditha | அநிஂதிதா
Beautiful, Virtuous, Venerated
Aninditha | அநிஂதிதா
Male
Gaelic
Variant form of Gaelic Cainneach, COINNEACH means "comely; finely made."
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Girl/Female
Tamil
Anindita | அநிஂதிதா
Beautiful, Virtuous, Venerated
Anindita | அநிஂதிதா
Boy/Male
Indian
Honored, Venerated
Girl/Female
American, Australian, Chinese, Scottish
Son of the Fair One; Fair Skinned; Comely; Finely Made
Girl/Female
Indian
Who is to be Venerated and Respected
Girl/Female
Hindu
Generates harmony in dance and music
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Boy/Male
Hindu, Indian
Self Generated; Lord Shiva
Boy/Male
Indian, Telugu
Generated
FINITELY GENERATED-MODULE
FINITELY GENERATED-MODULE
Boy/Male
Tamil
Ornamented, Lord Krishna
Boy/Male
Christian, German, Norse, Polish, Scandinavian, Swedish
Peaceful Ruler; Forever; Alone; Ruler; All-ruler
Girl/Female
Muslim/Islamic
Lioness
Boy/Male
Irish
Choice of the sea.
Girl/Female
Arabic, Muslim
Blossom; Flower
Girl/Female
Hindu
Beautiful
Girl/Female
Australian, Chinese, Danish, Dutch, Finnish, French, German, Hebrew, Netherlands, Polish, Slovenia, Swedish, Swiss
Full of Grace; God is Gracious; God has Shown Favor; Form for Anna
Female
English
Variant spelling of English Letitia, LATISHA means "happiness."
Girl/Female
Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Successful Lady; Victorious; Famous
Girl/Female
German
FINITELY GENERATED-MODULE
FINITELY GENERATED-MODULE
FINITELY GENERATED-MODULE
FINITELY GENERATED-MODULE
FINITELY GENERATED-MODULE
v. t.
To beget; to procreate; to propagate; to produce (a being similar to the parent); to engender; as, every animal generates its own species.
v. t.
To generate; to produce.
n.
That which generates.
p. pr. & vb. n.
of Generate
a.
Self-generated; produced independently.
imp. & p. p.
of Generate
a.
Generated by water.
a.
First formed or generated; original; primigenial.
a.
Relating to autogenesis; self-generated.
imp. & p. p.
of Venerate
adv.
Infinitely.
n.
One who, or that which, generates, begets, causes, or produces.
v. t.
To regard with reverential respect; to honor with mingled respect and awe; to reverence; to revere; as, we venerate parents and elders.
a.
Not begot; not yet generated; also, having never been generated; self-existent; eternal.
n.
Capability of being generated.
a.
Having a limit; limited in quantity, degree, or capacity; bounded; -- opposed to infinite; as, finite number; finite existence; a finite being; a finite mind; finite duration.
a.
Capable of being generated or produced.
adv.
In a finite manner or degree.
adv.
Without bounds or limits; beyond or below assignable limits; as, an infinitely large or infinitely small quantity.