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Direct summand of a free module (mathematics)
from free to projective modules: a module P is projective if and only if for every surjective module homomorphism f : N ↠ M and every module homomorphism
Projective_module
Algebraic structure in ring theory
algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over a ring
Flat_module
algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where
Kaplansky's theorem on projective modules
Kaplansky's_theorem_on_projective_modules
Generalization of vector spaces from fields to rings
R. These are the modules that behave very much like vector spaces. Projective Projective modules are direct summands of free modules and share many of
Module_(mathematics)
In mathematics, a module that has a basis
Kaplansky's theorem states a projective module over a (possibly non-commutative) local ring is free. Sometimes, whether a module is free or not is undecidable
Free_module
Mathematical object in abstract algebra
homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in (Baer 1940) and are discussed in some
Injective_module
Type of object in category theory
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used
Projective_object
a direct summand of free modules. In particular, every free module is projective. 2. The projective dimension of a module is the minimal length of (if
Glossary_of_module_theory
Exact sequence used to describe the structure of an object
example, a module has projective dimension zero if and only if it is a projective module. If M {\displaystyle M} does not admit a finite projective resolution
Resolution_(algebra)
In algebra, module with a finite generating set
equivalent conditions on a module. Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and
Finitely_generated_module
Relates the geometric vector bundles to algebraic projective modules
the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules are like vector bundles"
Serre–Swan_theorem
Commutative algebra theorem
commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement
Quillen–Suslin_theorem
Branch of algebra
\mathbf {P} (R)} the set of isomorphism classes of finitely generated projective modules over R; let also P n ( R ) {\displaystyle \mathbf {P} _{n}(R)} subsets
Ring_theory
Studies linear representations of finite groups over fields of positive characteristic
as non-isomorphic projective indecomposables have non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the
Modular_representation_theory
category theory, a projective cover of an object M is in a sense the best approximation of M by a projective object P. Projective covers are the dual
Projective_cover
Algebraic structure
research. Projective modules can be defined to be the direct summands of free modules. If R is local, any finitely generated projective module is actually
Commutative_ring
Topics referred to by the same term
variety Projective linear group Projective module Projective line Projective object Projective transformation Projective hierarchy Projective connection
Projective
Completion of the usual space with "points at infinity"
concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus
Projective_space
Algebra with unique prime factorization
submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is
Dedekind_domain
Statement in abstract algebra
{\displaystyle R^{n}} for a positive integer n. Since every free module is a projective module, there exists right inverse of the projection map (it suffices
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Pure injective and pure projective modules follow closely from the ideas of Prüfer's 1923 paper. While pure projective modules have not found as many applications
Pure_subgroup
dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However
Dualizing_module
On polynomial rings over fields
length k ≤ n. This upper bound on the projective dimension is sharp, that is, there are modules of projective dimension exactly n. The standard example
Hilbert's_syzygy_theorem
Module generated by a countable subset
that a projective module is a direct sum of countably generated modules. More generally, a module over a possibly non-commutative ring is projective if and
Countably_generated_module
Dedekind domain, a finitely generated torsion-free module is projective and a projective module is reflexive (the existence of a dual basis). Bourbaki, Nicolas
Torsionless_module
especially its simple modules, projective modules, and indecomposable modules. A (left) principal indecomposable module of a ring R is a (left) submodule
Principal indecomposable module
Principal_indecomposable_module
Ring whose ideals are projective
abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required
Hereditary_ring
Direct sum of irreducible modules
f(A)\oplus s(C).} In particular, any module over a semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple ring is
Semisimple_module
Module over a ring
particular free and projective modules, are torsion-free, but the converse need not be true. An example of a torsion-free module that is not flat is the
Torsion-free_module
f − g factors through a projective module. The stable module category is defined by setting the objects to be the R-modules, and the morphisms are the
Stable_module_category
Branch of mathematics
{\displaystyle K(\mathbb {P} ^{n})} for projective space over a field. This is because the intersection numbers of a projective X {\displaystyle X} can be computed
K-theory
Subject area in mathematics
group K(X) is now known as K0(X). Upon replacing vector bundles by projective modules, K0 also became defined for non-commutative rings, where it had applications
Algebraic_K-theory
Mathematical terminology
and sufficient for OL to be a projective module over Z[G]. It is certainly therefore necessary for it to be a free module. It leaves the question of the
Galois_representation
Branch of mathematics
When E {\displaystyle E} is a finitely generated projective module, it plays the role of the module of sections of a vector bundle. In the setting of
Noncommutative_geometry
Vector bundle of rank 1
{\displaystyle L} . In this way, projective space acquires a universal property. The universal way to determine a map to projective space is to map to the projectivization
Line_bundle
Operation that pairs a left and a right R-module into an abelian group
general modules, and the homomorphisms are isomorphisms if the modules E and F are finitely generated projective modules (in particular, free modules of finite
Tensor_product_of_modules
is quasi-isomorphic to a bounded complex of finite projective A-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated
Perfect_complex
French mathematician (1904–2008)
theory. They introduced fundamental concepts, including those of projective module, weak dimension, and what is now called the Cartan–Eilenberg resolution
Henri_Cartan
is a free module. A projective module is stably free if and only if it possesses a finite free resolution. An infinitely generated module is stably free
Stably_free_module
Algebraic object with geometric applications
generated projective modules is treated. The global sections of sections of a vector bundle over a compact space form a projective module over the ring
Tensor
Component of the Apollo spacecraft
The Apollo command and service module (CSM) was one of two principal components of the United States Apollo spacecraft, used for the Apollo program, which
Apollo command and service module
Apollo_command_and_service_module
Algebra associated to any vector space
projective geometry" A compilation of English translations of three notes by Cesare Burali-Forti on the application of exterior algebra to projective
Exterior_algebra
category of modules: Applications, Reinhard Fischer, ISBN 978-3889270177 Miyashita, Y. (1966), "Quasi-projective modules, perfect modules, and a theorem
Uniform_module
French mathematician (born 1926)
geometry. In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free. This question led to a great deal
Jean-Pierre_Serre
In algebra, integer associated to a module
In algebra, the length of a module over a ring R {\displaystyle R} is a generalization of the dimension of a vector space which measures its size. page
Length_of_a_module
Polish-American mathematician (1913–1998)
telescope) is a construction applying the telescoping cancellation idea to projective modules. Eilenberg contributed to automata theory and algebraic automata theory
Samuel_Eilenberg
Modular translation unit in C++
module must be declared using the word module to indicate that the translation unit is a module. A module, once compiled, is stored as a built module
Modules_(C++)
direct sum of fractional ideals. Every lattice over a Dedekind domain is projective. Lattices are well-behaved under localization and completion: A lattice
Lattice_(module)
Mathematical group formed from the automorphisms of an object
the field extension. The automorphism group of the projective n-space over a field k is the projective linear group PGL n ( k ) . {\displaystyle \operatorname
Automorphism_group
Theorem in algebra mathematics
algebraic geometry: Let f : X → Y {\textstyle f:X\to Y} be a projective morphism between quasi-projective varieties. Then f {\textstyle f} is an isomorphism if
Nakayama's_lemma
Ring that is also a vector space or a module
is an Ae-module by (x ⊗ y) ⋅ (a ⊗ b) = ax ⊗ yb. Equivalently, A is separable if it is a projective module over Ae; thus, the Ae-projective dimension
Associative_algebra
Branch of mathematics that studies algebraic structures
theory) Simple module, Semisimple module Indecomposable module Artinian module, Noetherian module Homological types: Projective module Projective cover Swan's
List of abstract algebra topics
List_of_abstract_algebra_topics
Class of crewed spacecraft from China
single-use vehicle composed of three modules; a descent module housing the crew during launch and reentry, an orbital module which provides additional living
Shenzhou_(spacecraft)
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
Planned lander spacecraft component of NASA's cancelled Project Constellation
The Altair spacecraft, previously known as the Lunar Surface Access Module or LSAM, was the planned lander spacecraft component of NASA's cancelled Constellation
Altair_(spacecraft)
a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but
Fake_projective_plane
Generalization of vector bundles
{\mathcal {O}}(1)} means that projective space is a Fano variety. Over the complex numbers, this means that projective space has a Kähler metric with
Coherent_sheaf
(Mathematical) ring with a unique maximal ideal
theorem by Irving Kaplansky says that any projective module over a local ring is free, though the case where the module is finitely-generated is a simple corollary
Local_ring
Series of spacecraft designed for the Soviet space programme
of three main sections. The descent module is where cosmonauts are seated for launch and reentry. The orbital module provides additional living space and
Soyuz_(spacecraft)
Type of integral domain
is coprime to w and R is a UFD and either every finitely generated projective module over R is free or c is 1 mod w, the ring R[X1, ..., Xn, Z]/(Zc − F(X1
Unique_factorization_domain
Projective construction in ring theory
mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A (with 1), the projective line P1(A)
Projective_line_over_a_ring
Assembly of photovoltaic cells used to generate electricity
device that converts sunlight into electricity by using multiple solar modules that consists of photovoltaic (PV) cells. PV cells are made of materials
Solar_panel
Invariant of rings and modules
commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension by the Auslander–Buchsbaum formula. A more elementary
Depth_(ring_theory)
Question in abstract algebra
C, then it is known that this is equivalent to A being free. (See Projective module). Caution: The converse of Whitehead's problem, namely that every
Whitehead_problem
Method of proof involving paradoxical properties of infinite sums
there is a free module F with A ⊕ F ≅ F. To see this, choose a module B such that A ⊕ B is free, which can be done as A is projective, and put F = B ⊕
Eilenberg–Mazur_swindle
point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval [0,1]
Lift_(mathematics)
Topic in abstract algebra
other words it is a quotient of a projective module by a projective submodule. Ext1 A(T,T ) = 0. The right A-module A is the kernel of a surjective morphism
Tilting_theory
Equivalence relation on rings
to S-Mod, then the R module M has any of the following properties if and only if the S module F(M) does: injective, projective, flat, faithful, simple
Morita_equivalence
Category whose objects are R-modules and whose morphisms are module homomorphisms
category of modules over some ring. Projective limits and inductive limits exist in the categories of left and right modules. Over a commutative ring, together
Category_of_modules
commutative ring assigned to any projective variety. If V is an algebraic variety given as a subvariety of projective space of a given dimension N, its
Homogeneous_coordinate_ring
American mathematician
theory established the concept of a relatively-projective module and explained its role in the theory of module decompositions. He developed a characterization
Donald_G._Higman
algebra is an algebra such that each left or right indecomposable projective module has a unique composition series. They were studied by Tadasi Nakayama (1940)
Nakayama_algebra
Type of sheaf of modules
one projective module over R. For example, this includes fractional ideals of algebraic number fields, since these are rank one projective modules over
Invertible_sheaf
Le Corbusier's anthropometric scale of proportions
The Modulor is an anthropometric scale of proportions devised by the Swiss-born French architect Le Corbusier (1887–1965). It was developed as a visual
Modulor
Algebraic formula
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 ) = d e p t h (
Auslander–Buchsbaum_formula
Soviet/Russian space station (1986–2001)
7.5-tonne (8.3-short-ton) modules derived from the Soyuz spacecraft. These modules would have used a Soyuz propulsion module, as used in Soyuz and Progress;
Mir
the "smaller" projective generators Using the characterisation in terms of projections from coproducts, having a family of projective generators along
Generator_(category_theory)
Would relate vector bundles over a regular Noetherian ring and over a polynomial ring
conjecture. The conjecture is a statement about finitely generated projective modules. Such modules are also referred to as vector bundles. For a ring A, the set
Bass–Quillen_conjecture
Projective analogue of the spectrum of a ring
schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental
Proj_construction
R-module has a projective cover. Every finitely generated left (right) R-module has a projective cover. The category of finitely generated projective R
Perfect_ring
Topics referred to by the same term
Look up module or modular in Wiktionary, the free dictionary. Module, modular and modularity may refer to the concept of modularity. They may also refer
Module
Mathematical object studied in the field of algebraic geometry
plays an important role in the theory of D-modules. A projective variety is a closed subvariety of a projective space. That is, it is the zero locus of a
Algebraic_variety
Generalization of complex inner products
and the twist is provided by a field automorphism. An application in projective geometry requires that the scalars come from a division ring (skew field)
Sesquilinear_form
Real numbers adjoined with a nil-squaring element
points of the projective line over D are equivalence classes in B under this relation: P(D) = B/~. They are represented with projective coordinates [a
Dual_number
Group of unitary matrices
as subgroup and the projective orthogonal group PO ( n ) {\displaystyle \operatorname {PO} (n)} as quotient, and the projective special orthogonal group
Unitary_group
Sheaf consisting of modules on a ringed space; generalizing vector bundles
an OX-module in a natural way. O-modules arising in such a fashion are examples of quasi-coherent sheaves, and in fact, on affine or projective schemes
Sheaf_of_modules
Module components with flexibility in module theory
remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct
Pure_submodule
map induces a projectivity. The converse of this observation (except for the projective line) is the fundamental theorem of projective geometry. Thus
Semilinear_map
Endomorphism algebra of an abelian group
endomorphism ring of a continuous module or discrete module is a clean ring. If an R module is finitely generated and projective (that is, a progenerator), then
Endomorphism_ring
First international crewed spaceflight mission
leftover from the canceled Apollo missions program and was the final Apollo module to fly. The crew consisted of American astronauts Thomas P. Stafford, Vance
Apollo–Soyuz
formed a one-step projective resolution of a module, and remarked that if the kernel was projective in one resolution it was projective in all. I added
Schanuel's_lemma
Concept in abstract algebra
{\displaystyle RG} -module R has a projective resolution of length n, i.e. there are projective R G {\displaystyle RG} -modules P 0 , … , P n {\displaystyle
Cohomological_dimension
projective 1. A projective module is a module such that every epimorphism to it splits. 2. A projective resolution is a resolution by projective modules
Glossary of commutative algebra
Glossary_of_commutative_algebra
Generalizes showing that two homology theories are isomorphic
free functor F {\displaystyle F} is basically just a free (and hence projective) module. V {\displaystyle V} being acyclic at the models (there is only one)
Acyclic_model
Space with one dimension
correspondence. More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the
One-dimensional_space
regular ring. (C) The algebra K[V] is a free module over K[V]G. (C') The algebra K[V] is a projective module over K[V]G. In the case when K is the field
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Fourth crewed Moon landing
Command Module Pilot Alfred Worden orbited the Moon, operating the sensors in the scientific instrument module (SIM) bay of the service module. This suite
Apollo_15
*-algebra of bounded operators on a Hilbert space
semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts to axiomatize the underlying
Von_Neumann_algebra
Construction in homological algebra
i > 0 {\displaystyle i>0} if the R {\displaystyle R} -module A {\displaystyle A} is projective (for example, free) or if B {\displaystyle B} is injective
Ext_functor
Topics referred to by the same term
subsets Countably generated module. (Kaplansky's theorem says that a projective module is a direct sum of countably generated modules.) This disambiguation
Countably_generated
PROJECTIVE MODULE
PROJECTIVE MODULE
Boy/Male
Polish
Protective shield.
Girl/Female
Irish
Protective.
Girl/Female
German, Italian, Swedish
Protective; Victorious Shield
Girl/Female
Muslim
Protective Angel
Boy/Male
Christian & English(British/American/Australian)
Protective Friend
Boy/Male
Arabic, Indian, Muslim, Sindhi
Protective; Safety
Boy/Male
British, English, Netherlands
Protective
Girl/Female
Indian
Protective Angel
Girl/Female
German American
Protective.
Girl/Female
German, Swedish
Protective Victory
Boy/Male
Christian & English(British/American/Australian)
Protective Grace
Girl/Female
Celtic, French, German, Irish
Strong; Protective
Girl/Female
Muslim
Protective Angel
Girl/Female
Muslim/Islamic
Protective angel
Girl/Female
Irish
Protective.
Boy/Male
Greek
Productive.
Boy/Male
German
Protective
Girl/Female
Indian
Protective Angel
Girl/Female
Muslim/Islamic
Protective angel
Boy/Male
German
Protective
PROJECTIVE MODULE
PROJECTIVE MODULE
Boy/Male
Hindu, Indian
Lord Krishna
Girl/Female
Latin
Beautiful Christian.
Boy/Male
Hindu
One who attracts everything in the world to him
Girl/Female
Australian, British, Christian, English, Hebrew
God Sees
Girl/Female
Egyptian
Daughter of the Nile.
Boy/Male
Tamil
A cavalier, A Hindu month, Medical God
Female
Gypsy/Romani
(Лала) Bulgarian name LALA means "tulip." In use by the Romani. Compare with other forms of Lala.
Surname or Lastname
English
English : from a pet form of Fulcher.German (also Füge) : nickname for a skillful, adroit person, from Middle High German vüege ‘skillful’, ‘fitting’ (see Fiegel).
Boy/Male
Hindu, Indian
A Person who has No Enemy
Boy/Male
Hebrew
Saved from the water.
PROJECTIVE MODULE
PROJECTIVE MODULE
PROJECTIVE MODULE
PROJECTIVE MODULE
PROJECTIVE MODULE
a.
Caused or imparted by impulse or projection; impelled forward; as, projectile motion.
n.
Of or pertaining to a prospect; furnishing a prospect; perspective.
a.
Projecting or impelling forward; as, a projectile force.
n.
A jutting out; also, a part jutting out, as of a building; an extension beyond something else.
n.
Looking forward in time; acting with foresight; -- opposed to retrospective.
n.
A perspective glass.
n.
Being within view or consideration, as a future event or contingency; relating to the future: expected; as, a prospective benefit.
n.
Any method of representing the surface of the earth upon a plane.
n.
The act of scheming or planning; also, that which is planned; contrivance; design; plan.
a.
Affording protection; sheltering; defensive.
a.
Pertaining to projection, or to a projectile.
n.
A part of mechanics which treats of the motion, range, time of flight, etc., of bodies thrown or driven through the air by an impelling force.
n.
The quality or state of projecting, or being projected; projection; protrusion.
n.
The act of throwing or shooting forward.
n.
The scene before or around, in time or in space; view; prospect.
n.
A jutting out beyond a surface.
n.
A body projected, or impelled forward, by force; especially, a missile adapted to be shot from a firearm.
a.
Having the quality or power of producing; yielding or furnishing results; as, productive soil; productive enterprises; productive labor, that which increases the number or amount of products.
a.
Bringing into being; causing to exist; producing; originative; as, an age productive of great men; a spirit productive of heroic achievements.
n.
The representation of something; delineation; plan; especially, the representation of any object on a perspective plane, or such a delineation as would result were the chief points of the object thrown forward upon the plane, each in the direction of a line drawn through it from a given point of sight, or central point; as, the projection of a sphere. The several kinds of projection differ according to the assumed point of sight and plane of projection in each.