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Module generated by a countable subset
mathematics, a module over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance
Countably_generated_module
In algebra, module with a finite generating set
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, finite
Finitely_generated_module
Topics referred to by the same term
generating set Countably generated space, a topological space in which the topology is determined by its countable subsets Countably generated module. (Kaplansky's
Countably_generated
Direct summand of a free module (mathematics)
a direct product of countably many copies of F2 and I is the direct sum of countably many copies of F2 inside of R. The R-module R/I is locally free since
Projective_module
M_{i},i\in I} are countably generated modules with local endomorphism rings and if N {\displaystyle N} is a countably generated module that is a direct
Kaplansky's theorem on projective modules
Kaplansky's_theorem_on_projective_modules
Concept in mathematics
a module. Countably generated module Flat module Invariant basis number "ac.commutative algebra – Existence of a minimal generating set of a module –
Generating_set_of_a_module
convergent sequences. The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well
Countably_generated_space
continuous module countably generated A countably generated module is a module that admits a generating set whose cardinality is at most countable. cyclic
Glossary_of_module_theory
Type of algebra
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
Mathematical objects that generalise the notion of Hilbert spaces
_{C(X)}(x):=g(\sigma (x),\rho (x)).} The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra A = C ( X ) {\displaystyle
Hilbert_C*-module
Commutative group where every element is the sum of elements from one finite subset
of cyclic groups. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers
Finitely generated abelian group
Finitely_generated_abelian_group
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
modules is torsionless. A free module is reflexive if it is finitely generated, and for some rings there are also infinitely generated free modules that
Torsionless_module
Abstract algebra concept
decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example
Decomposition_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 ideal
Abelian_group
*-algebra of bounded operators on a Hilbert space
Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts
Von_Neumann_algebra
Property of topological spaces
Hausdorff. Compact-open topology – Type of topology Countably generated space Finitely generated space – Type of topology in mathematicsPages displaying
Compactly_generated_space
Field extension that is not algebraic
separably generated if it admits a separating transcendence basis. If a field extension is finitely generated and it is also separably generated, then each
Transcendental_extension
Abelian group in which every element can, in some sense, be divided by positive integers
hereditary, so any submodule generated by injective modules is injective. The converse is a result of (Matlis 1958): if every module has a unique maximal injective
Divisible_group
Class of mathematical sets
empty set and the entire set X {\displaystyle X} , and is closed under countable union and complement. Then we can define the Borel σ-algebra over X {\displaystyle
Borel_set
Monster and modular connection
representation in n dimensions (which is itself isomorphic to a polynomial ring in countably infinitely many generators). For the case in question, one sets L to be
Monstrous_moonshine
Mathematical property of a space
every sequence has a convergent subsequence. Countably compact. A space is countably compact if every countable open cover has a finite subcover. Pseudocompact
Topological_property
Set of vectors used to define coordinates
retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles
Basis_(linear_algebra)
Algebra with unique prime factorization
finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind
Dedekind_domain
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
Function that returns cardinal numbers
t(X)=\aleph _{0}} the space X {\displaystyle X} is said to be countably generated or countably tight. The augmented tightness of a space X , {\displaystyle
Cardinal_function
Algebra of formal sums
may equivalently be called free Z {\displaystyle \mathbb {Z} } -modules, the free modules over the integers. Lattice theory studies free abelian subgroups
Free_abelian_group
Set of finitely supported functions from a group to a ring
ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars
Group_ring
Structure in Ring Theory (Mathematics)
Nakayama's lemma. This lemma is a technical tool for studying finitely generated modules over commutative rings that has an easy geometric interpretation:
Jacobson_radical
A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological
Persistence_module
Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable space Lindelöf
List of general topology topics
List_of_general_topology_topics
Mathematical group that can be generated as the set of powers of a single element
finitely generated abelian group or nilpotent group is polycyclic. Cycle graph (group) Cyclic module Cyclic sieving Prüfer group (countably infinite analogue)
Cyclic_group
Ideal in a ring which has properties similar to prime elements
In a commutative ring, an ideal maximal with respect to being not countably generated is prime. Radical ideal Maximal ideal Dedekind–Kummer theorem Residue
Prime_ideal
Topological space that is homeomorphic to a metric space
σ-locally finite base. A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets. For a closely related theorem
Metrizable_space
the Jacobson radical. Every finitely generated algebra over R that is a field is finitely generated as an R-module. (Zariski's lemma) Every prime ideal
Jacobson_ring
R has the invariant basis number (IBN) property if all finitely generated free modules over R have a well-defined rank. In the case of fields, the IBN
Invariant_basis_number
Theory in mathematics
cycles is the set of triples (H, ρ, F), where H is a countably generated graded Hilbert module over B, ρ is a *-representation of A on H as even bounded
KK-theory
Specification of a mathematical group by generators and relations
this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that
Presentation_of_a_group
analytic functions which takes at most countably many distinct values at every point is necessarily countable, is true if and only if the continuum hypothesis
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Function from sets to numbers
In particular, this is why the definition of "countably additive" is rarely extended from countably many sets F 1 , F 2 , … {\displaystyle F_{1},F_{2}
Set_function
Sporadic simple group
have been completely classified. Every such group belongs to one of 18 countably infinite families or is one of 26 sporadic groups that do not follow such
Monster_group
Set with operations obeying given axioms
algebra is an algebraic structure that is a vector space over a field or a module over a commutative ring. The collection of all structures of a given type
Algebraic_structure
Algebraic structure formed from a collection of algebraic structures
abelian groups, vector spaces, or modules. For example, consider the direct sum and the direct product of (countably) infinitely many copies of the integers
Direct_sum
while the exp ring is a polynomial ring in countably many generators. For each element g of G introduce a countable set of variables gi for i>0. Define exp(gt)
Exp_algebra
. . . ] ] {\displaystyle R[[X_{1},X_{2},...]]} over R in infinitely (countably) many indeterminates; the elements of this power series ring are formal
Ring_of_symmetric_functions
Subgroup of an abelian group consisting of all elements of finite order
torsion-free if and only if it is flat as a Z {\displaystyle \mathbb {Z} } -module, which means that whenever C {\displaystyle C} is a subgroup of some abelian
Torsion_subgroup
Property of subsets of ordered vector spaces
vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum. Being an
Order_complete
Topology on Cartesian products of topological spaces
\mathbb {R} ^{n}.} ) The Cantor set is homeomorphic to the product of countably many copies of the discrete space { 0 , 1 } {\displaystyle \{0,1\}} and
Product_topology
Isomorphism of commutative rings constructed in the theory of Lie algebras
{g}})} and its center acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vector).
Harish-Chandra_isomorphism
Mathematical term in group theory
which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian
Prüfer_group
families of the Ind-Pro objects are countable) are equivalent to countably generated Tate R-modules in the sense of Drinfeld mentioned above. A Tate Lie algebra
Tate_vector_space
Study of discrete mathematical structures
finite or countably infinite sets. Hopkins, Brian, ed. (2009). Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles
Discrete_mathematics
Mathematical group formed from the automorphisms of an object
\operatorname {Aut} (B)} (cf. #In category theory). Let P be a finitely generated projective module over a ring R. Then there is an embedding Aut ( P ) ↪ GL n
Automorphism_group
Type of vector space in math
with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the
Hilbert_space
Topological space that is connected
commutative ring R {\displaystyle R} is connected Every finitely generated projective module over R {\displaystyle R} has constant rank. R {\displaystyle
Connected_space
example of a countably infinite Hausdorff space that is connected. The third topology, introduced by A.M. Kirch, is an example of a countably infinite Hausdorff
Arithmetic progression topologies
Arithmetic_progression_topologies
Maximal proper filter
intersection of any countable collection of elements of U {\displaystyle U} is still in U {\displaystyle U} —is called countably complete or σ-complete
Ultrafilter_on_a_set
Measure theory and probability theorem
M {\displaystyle M} of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M {\displaystyle
Monotone_class_theorem
Czech mathematician
*-module is finitely generated", Journal of Algebra, 169 (2): 392–398, doi:10.1006/jabr.1994.1291 1996: Trlifaj, Jan (1996), "Whitehead test modules",
Jan_Trlifaj
be the exterior algebra over a countably infinite-dimensional vector space with basis e1, e2, e3, ... Then R is generated by the elements of this basis
Polynomial_identity_ring
Family closed under subsets and countable unions
contain measurable subsets and countable unions of its elements. The concept of 𝜎-ideal is dual to that of a countably complete (𝜎-) filter. If a measure
Sigma-ideal
Direct limit of a direct system of groups
with direct limit (isomorphic to) the subgroup of the symmetric group on countably many things S ω {\displaystyle S_{\omega }} which contains permutations
Direct_limit_of_groups
Algebra in statistical mechanics
Temperley-Lieb algebra. The cell module W ℓ , z {\displaystyle W_{\ell ,z}} of a T L n ( δ ) {\displaystyle aTL_{n}(\delta )} is generated by the set of monic pairings
Temperley–Lieb_algebra
Important problem in lattice theory
join of the finitely generated congruences below it (e.g., every submodule of a module is the union of all its finitely generated submodules), we obtain
Congruence_lattice_problem
Mathematical construction relating to infinite-dimensional spaces
{\displaystyle \mu } does not extend to a countably additive measure on the σ {\displaystyle \sigma } -algebra generated by the collection of cylinder sets in
Abstract_Wiener_space
Class of algebraic structures
form a variety, as an arbitrary product of finitely generated abelian groups is not finitely generated. Viewing a variety V and its homomorphisms as a category
Variety_(universal_algebra)
Type of group in abstract algebra
one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S5 are generated by elements like (123)(45) that have
Symmetric_group
Vectorizing features using a hash function
Mathematically, a token is an element t {\displaystyle t} in a finite (or countably infinite) set T {\displaystyle T} . Suppose we only need to process a
Feature_hashing
Subset with finite complement
has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an infinite
Cofiniteness
-modules over some (unitary, associative) ring R {\displaystyle R} , the finitely presentable objects are precisely the finitely presented modules. The
Accessible_category
Foundations of probability theory
general relax the third axiom. In order to demonstrate that the theory generated by the Kolmogorov axioms corresponds with classical probability, some
Probability_axioms
Special subset of a partially ordered set
linear map f : A → B. Given any infinite cardinal κ, the modules in I that cannot be generated by fewer than κ elements form a filter. Every uniform structure
Filter_(mathematics)
Scripting language created in 1994
usually processed on a web server by a PHP interpreter implemented as a module, a daemon or a Common Gateway Interface (CGI) executable. On a web server
PHP
Mathematical concept
\operatorname {End} (V)} of a vector space (or free module) V {\displaystyle V} with a countably infinite basis e 1 , e 2 , … {\displaystyle e_{1},e_{2}
Dedekind-finite_ring
Tool in algebraic topology
Hj(U,E) → Hj(V,E) is a finitely generated R-module. Then the cohomology groups Hj(X,E) are finitely generated R-modules. For example, for a compact Hausdorff
Sheaf_cohomology
Field theory involving topological effects in physics
projective space, and if such a thing could be defined it would have countably infinitely many degrees of freedom.) The known topological field theories
Topological quantum field theory
Topological_quantum_field_theory
Duality for locally compact abelian groups
σ-algebra generated by the compact sets. More precisely, a right Haar measure on a locally compact group G {\displaystyle G} is a countably additive measure
Pontryagin_duality
Programming language
upgrading–patching module packages. c2nim is a source-to-source compiler (transcompiler or transpiler) meant to be used on C/C++ headers to help generate new Nim
Nim_(programming_language)
Branch of mathematics
one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "map" is
Homotopy_theory
Programming language
entities inside the module. By inserting import ... statements between module/program declaration and definition areas, modules can share their own entities
Pascal_(programming_language)
Family closed under complements and countable disjoint unions
{\displaystyle B\setminus A\in D;} D {\displaystyle D} is closed under countable increasing unions: if A 1 ⊆ A 2 ⊆ A 3 ⊆ ⋯ {\displaystyle A_{1}\subseteq
Dynkin_system
Measure space in mathematics
ones preventing completeness from holding true); let Σ0 be the σ-algebra generated by Σ and Z (i.e. the smallest σ-algebra that contains every element of
Complete_measure
Algebraic concept in measure theory, also referred to as an algebra of sets
sets. The Loomis-Sikorski theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called abstract sigma algebras)
Field_of_sets
represented in analytic geometry and it is generalized in operator theory and in module theory. Sometimes matrix theory is considered a branch, although linear
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Topological space construction
{\displaystyle \{\mathbb {Z} \}\cup (\mathbb {R} \setminus \mathbb {Z} )} ) is a countably infinite bouquet of circles joined at a single point Z . {\displaystyle
Quotient_space_(topology)
Algebraic ring that need not have additive negative elements
of (isomorphism equivalence classes of) combinatorial classes (sets of countably many objects with non-negative integer sizes such that there are finitely
Semiring
Concept in mathematics
{\mathcal {R}}} , may be viewed as the space of observables of the system of countably infinite number of distinct spin 1 / 2 {\displaystyle 1/2} fermions. Each
Walsh_function
2^{\aleph _{0}}} . Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ℵ ω 1 {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
In mathematics, vector space of linear forms
\mathbb {R} ^{\infty }} is countably infinite, whereas R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} does not have a countable basis. This observation
Dual_space
Declarative logic programming language
Datalog inference capabilities. Could be used as httpd (Apache HTTP Server) module or standalone (although beta versions are under the Perl Artistic License
Datalog
Graph generated by a random process
vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random
Random_graph
straightforward extension of the fundamental theorem of finitely generated abelian groups to countable abelian p-groups without elements of infinite height: each
Height_(abelian_group)
Group with translationally invariant total order
doi:10.1007/BF03174799, S2CID 198139979 Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84
Linearly_ordered_group
Method of deriving conclusions
metavariables—placeholders that can be replaced by specific terms or formulas to generate an infinite number of true statements. For example, propositional logic
Rule_of_inference
Algebraic structure in linear algebra
occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the
Vector_space
Two closely related models for generating random graphs
Infinite graph containing all countable graphs, the graph formed by extending the G(n, p) model to graphs with a countably infinite number of vertices.
Erdős–Rényi_model
Relation among continuous functions
the change in functions More generally, on any compactly generated space; e.g., a first-countable space. Rudin 1991, p. 44 §2.5. Reed & Simon (1980), p.
Equicontinuity
Space with topology generated by convex sets
space ∏ i ∈ N R {\textstyle \prod _{i\in \mathbb {N} }\mathbb {R} } of countably many copies of R {\displaystyle \mathbb {R} } (this homeomorphism need
Locally convex topological vector space
Locally_convex_topological_vector_space
Identifies the commutant of a specific von Neumann algebra
defines a tracial state on M. It is called cyclic since Ω generates H as a topological M-module. It is called separating because if aΩ = 0 for a in M, then
Commutation theorem for traces
Commutation_theorem_for_traces
Algebraic variety defined within an affine space
algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime. One-dimensional affine varieties
Affine_variety
COUNTABLY GENERATED-MODULE
COUNTABLY GENERATED-MODULE
Girl/Female
Tamil
Aninditha | அநிஂதிதா
Beautiful, Virtuous, Venerated
Aninditha | அநிஂதிதா
Girl/Female
French
From the countly estate.
Boy/Male
Indian, Sanskrit
Devoted; Venerated
Boy/Male
Indian
Honored, Venerated
Girl/Female
Tamil
Generates harmony in dance and music
Boy/Male
Hindu, Indian
Un Countable; Multiple; Countless
Boy/Male
Hindu
Generator, Producer, Father (King of Mithila; Father of Sita, who found her in a furrow)
Boy/Male
Arabic, Muslim, Sindhi
Venerated; Honoured
Girl/Female
Indian
Who is to be Venerated and Respected
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Biblical
penetrated
Girl/Female
French
From the countly estate.
Boy/Male
Muslim
Honored, Venerated
Girl/Female
Tamil
Anindita | அநிஂதிதா
Beautiful, Virtuous, Venerated
Anindita | அநிஂதிதா
Girl/Female
Biblical
Penetrated.
Boy/Male
Hindu, Indian
Self Generated; Lord Shiva
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Sindhi, Tamil, Telugu
Virtuous; Venerated
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Girl/Female
Hindu
Generates harmony in dance and music
Boy/Male
Indian, Telugu
Generated
COUNTABLY GENERATED-MODULE
COUNTABLY GENERATED-MODULE
Girl/Female
American, Arabic, Australian, Chinese, French, Latin, Spanish
Valley of the Wolves; River of the Wolf
Girl/Female
Russian
Pearl.
Surname or Lastname
English
English : probably a variant of Pennywell.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Eternal Fame
Girl/Female
Hindu, Indian, Kannada
Virtuous; Full of Virtues
Boy/Male
Indian
The enricher, The emancipator
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Decorated with Flowers
Girl/Female
Hindu, Indian
Mighty Poet
Girl/Female
Hindu
Truth, Morality, Justice, Good behavior
Girl/Female
Muslim
Music Rhythm
COUNTABLY GENERATED-MODULE
COUNTABLY GENERATED-MODULE
COUNTABLY GENERATED-MODULE
COUNTABLY GENERATED-MODULE
COUNTABLY GENERATED-MODULE
a.
Self-generated; produced independently.
imp. & p. p.
of Venerate
a.
Not begot; not yet generated; also, having never been generated; self-existent; eternal.
v. t.
To generate; to produce.
v. t.
To regard with reverential respect; to honor with mingled respect and awe; to reverence; to revere; as, we venerate parents and elders.
n.
One who, or that which, generates, begets, causes, or produces.
n.
That which generates.
imp. & p. p.
of Generate
n.
Capability of being generated.
p. pr. & vb. n.
of Generate
a.
Relating to autogenesis; self-generated.
n.
The state or quality of being numerable or countable.
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.
a.
Capable of being numbered.
a.
First formed or generated; original; primigenial.
a.
Capable of being generated or produced.
v. t.
To produce or generate within.
a.
Generated by water.
n.
The place where anything is generated or produced.