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Mathematical object in abstract algebra
measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal
Injective_module
Notion in abstract algebra
particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential
Injective_hull
Pure-injective modules in mathematics
algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact
Algebraically_compact_module
Mathematical object in category theory
field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology
Injective_object
Mathematical ring with well-behaved ideals
if every direct sum of injective (left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct
Noetherian_ring
injective hull) is a maximal essential extension, or a minimal embedding in an injective module. 3. An injective cogenerator is an injective module such
Glossary_of_module_theory
Abelian group in which every element can, in some sense, be divided by positive integers
generated by injective modules is injective. The converse is a result of (Matlis 1958): if every module has a unique maximal injective submodule, then
Divisible_group
Exact sequence used to describe the structure of an object
projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a
Resolution_(algebra)
Generalization of vector spaces from fields to rings
many of their desirable properties. Injective Injective modules are defined dually to projective modules. Flat A module is called flat if taking the tensor
Module_(mathematics)
Properties of mathematical functions
g\circ f} is injective, then it can only be concluded that f {\displaystyle f} is injective (see figure). Every embedding is injective. A function is
Bijection, injection and surjection
Bijection,_injection_and_surjection
Algebraic structure in ring theory
_{R}S} is injective. Hence, M → M ⊗ R S {\displaystyle M\to M\otimes _{R}S} is injective. Conversely, if M ≠ 0 {\displaystyle M\neq 0} is a module over R
Flat_module
Construction of a ring of fractions
This implies that, if f : M → N {\displaystyle f\colon M\to N} is an injective module homomorphism, then S − 1 R ⊗ R f : S − 1 R ⊗ R M → S − 1 R ⊗ R N {\displaystyle
Localization (commutative algebra)
Localization_(commutative_algebra)
Prime ideal that is an annihilator of a prime submodule
coprimary modules. For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the
Associated_prime
{\displaystyle \Omega ^{-1}} can be defined as follows. Given M, find an injective module I with an inclusion i : M → I {\displaystyle i\colon M\to I} . Then
Stable_module_category
Direct summand of a free module (mathematics)
more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property
Projective_module
Concept in mathematics
necessarily an injective module, and is unique up to isomorphism. The injective hull is also minimal in the sense that any other injective module containing
Essential_extension
discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective. The group ( Q / Z , + ) {\displaystyle
Character_module
Bass (1963, p.11). The Bass numbers describe the minimal injective resolution of a finitely-generated module M over a Noetherian ring: for each prime ideal p
Bass_number
homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = Spec ( A ) {\displaystyle
Dual_module
Endomorphism algebra of an abelian group
If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring. For a semisimple module, the endomorphism
Endomorphism_ring
Homological algebra is the study of homological functors
Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective
List of homological algebra topics
List_of_homological_algebra_topics
dim(M) = n if and only if E(M) is a direct sum of n indecomposable injective modules. It can be shown that u.dim(M) = ∞ if and only if M contains an infinite
Uniform_module
Branch of mathematics that studies algebraic structures
theorem Injective module Injective hull Flat module Flat cover Coherent module Finitely-generated module Finitely-presented module Finitely related module Algebraically
List of abstract algebra topics
List_of_abstract_algebra_topics
envelope or injective hull of a module is a smallest injective module containing it. 3. An injective resolution is a resolution by injective modules. 4. The
Glossary of commutative algebra
Glossary_of_commutative_algebra
Linear map over a ring
algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring
Module_homomorphism
In algebra, module with a finite generating set
finitely generated modules. For example, if f : M → M is a surjective R-endomorphism of a finitely generated module M, then f is also injective, and hence is
Finitely_generated_module
NASA crewed Moon landing spacecraft (1969–1972)
The Apollo Lunar Module (LM /ˈlɛm/), originally designated the Lunar Excursion Module (LEM), was the lunar lander spacecraft that was flown between lunar
Apollo_Lunar_Module
Direct sum of irreducible modules
{\displaystyle B\cong f(A)\oplus s(C).} In particular, any module over a semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple
Semisimple_module
Theorem in algebra
residue field k, and choose E to be an injective hull of k (sometimes called a Matlis module). The dual DR(M) of a module M is defined to be HomR(M,E). Then
Matlis_duality
Abstract algebra concept
holds because every module is a quotient of a free module, and a quotient of a semisimple module is semisimple. Pure-injective module Anderson & Fuller
Decomposition_of_a_module
considered as a module over itself is a dualizing module. If R is an Artinian local ring then the Matlis module of R (the injective hull of the residue
Dualizing_module
Automotive control system
control module (ECM), powertrain control module (PCM), transmission control module (TCM), brake control module (BCM or EBCM), central control module (CCM)
Electronic_control_unit
Type of module over a ring
right) modules over the same ring, and let f : M → N be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because
Simple_module
Apollo Lunar Module rocket engine
Apollo Lunar Module descent stage. It used Aerozine 50 fuel and dinitrogen tetroxide (N 2O 4) oxidizer. This engine used a pintle injector, which paved
Descent_propulsion_system
Local ring in commutative algebra
ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed
Gorenstein_ring
Branch of mathematics
1956 book "Homological Algebra", these authors used projective and injective module resolutions. 'Tohoku': The approach in a celebrated paper by Alexander
Homological_algebra
Commutative group (mathematics)
G=A\oplus C} . Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's
Abelian_group
Overview of and topical guide to algebraic structures
values in F. Other special types of modules, including free modules, projective modules, injective modules and flat modules are studied in abstract algebra
Outline of algebraic structures
Outline_of_algebraic_structures
Concept in mathematics
In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing
Drinfeld_module
A particular algebraic structure
every simple R-module is injective. The following three conditions are equivalent: Every simple left (respectively right) R-module is injective. The radical
V-ring_(ring_theory)
D(M) of a left Λ-module M is the right Λ-module D(M) = HomR(M,J), where J is the dualizing module of R, equal to the sum of the injective envelopes of the
Artin_algebra
Mathematical object
elementary interaction with projective objects and injective objects. The two results are: An injective hopfian object is cohopfian. A projective cohopfian
Hopfian_object
Category with direct sums and certain types of kernels and cokernels
_{3}} identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of I is a full, additive
Abelian_category
Category whose objects are R-modules and whose morphisms are module homomorphisms
left modules over R {\displaystyle R} is the category whose objects are all left modules over R {\displaystyle R} and whose morphisms are all module homomorphisms
Category_of_modules
Apollo Lunar Module rocket engine
lunar module ascent engine (LMAE) is a fixed-thrust hypergolic rocket engine developed by Bell Aerosystems for use in the Apollo Lunar Module ascent
Ascent_propulsion_system
Type of object in category theory
the injective objects in R {\displaystyle R} -Mod are exactly the injective left R-modules. The category of left (right) R {\displaystyle R} -modules also
Projective_object
General concept and operation in mathematics
to a dual module. There is still a canonical evaluation map, but it is not always injective; if it is, this is known as a torsionless module; if it is
Duality_(mathematics)
deduced from the characterization of a Noetherian ring in terms of injective modules, as done for example by David Eisenbud in (Eisenbud 1970); this approach
Eakin–Nagata_theorem
Operation that pairs a left and a right R-module into an abelian group
of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction
Tensor_product_of_modules
self-injective A ring R is left self-injective if the module RR is an injective module. While rings with unity are always projective as modules, they
Glossary_of_ring_theory
case that a module M is simple, then it is necessarily the top of its projective cover, if it exists. The injective envelope for a module always exists
Projective_cover
Mathematical object in sheaf cohomology
example the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;
Injective_sheaf
Study of dimension in algebraic geometry
algebra. Let R {\displaystyle R} be a ring. The injective dimension of an R {\displaystyle R} -module M {\displaystyle M} denoted by id R M {\displaystyle
Dimension_theory_(algebra)
Index of articles associated with the same name
decomposes into the direct sum of its socle and cosocle.) Injective hull Radical of a module Cosocle Robinson 1996, p.87. J. L. Alperin; Rowen B. Bell
Socle_(mathematics)
Commutative algebra studies commutative rings, their ideals, and modules over such rings
ideal Hilbert's Nullstellensatz Flat module Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay ring Gorenstein ring Complete
List of commutative algebra topics
List_of_commutative_algebra_topics
Algebraic structure with "nice" duality properties
right self-injective. For a field k, a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right A-module Homk(A,k)
Frobenius_algebra
Construction in homological algebra
the R {\displaystyle R} -module A {\displaystyle A} is projective (for example, free) or if B {\displaystyle B} is injective. The converses also hold:
Ext_functor
Special subset of a partially ordered set
non-atomic partial orders forms a filter. Likewise, if I is the set of injective modules over some given commutative ring, of limited cardinality, modulo isomorphism
Filter_(mathematics)
Toolkit for generating malware
combines older rootkit source code with new functions for unpacking and injecting modules into user processes. Packed content is compressed using the LZ77 algorithm
BlackEnergy
German mathematician (1902–1979)
German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings, Baer groups, and Baer subplanes
Reinhold_Baer
Notion in metric geometry
as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the injective hull of
Tight_span
Module components with flexibility in module theory
the natural injective map. Then P {\displaystyle P} is a pure submodule of M {\displaystyle M} if, for any (right) R {\displaystyle R} -module X {\displaystyle
Pure_submodule
Index of articles associated with the same name
dimension of a module, based on projective resolutions Injective dimension of a module, based on injective resolutions Weak dimension of a module, or flat dimension
Homological_dimension
nonsingularity has a strong interaction with right self injective rings as well. Theorem: If R is a right self injective ring, then the following conditions on R are
Singular_submodule
identified within the injective hull. Let S=EndR(E(M)) be the endomorphism ring of the injective hull. Then an element x of the injective hull is in the rational
Dense_submodule
American mathematician (1923–2015)
for his contributions to the theory of rings and modules, especially for his work with injective modules over commutative Noetherian rings, and his introduction
Eben_Matlis
Homological construction in category theory
I^{0}\to I^{1}\to I^{2}\to \cdots } where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence
Derived_functor
First crewed Moon landing (1969)
mission was crewed by Commander Neil Armstrong, Command Module Pilot Michael Collins, and Lunar Module Pilot Edwin "Buzz" Aldrin, all of whom were on their
Apollo_11
Canadian mathematician (1922–2014)
earlier work was mostly in module theory, especially torsion theories, non-commutative localization, and injective modules. One of his earliest papers
Joachim_Lambek
f\in M^{\ast },} is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive
Torsionless_module
Sheaf consisting of modules on a ringed space; generalizing vector bundles
line bundle, some power of it is generated by global sections.) An injective O-module is flasque (i.e., all restrictions maps F(U) → F(V) are surjective)
Sheaf_of_modules
Structure-preserving map between two algebraic structures of the same type
then f {\displaystyle f} is bijective. In fact, f {\displaystyle f} is injective, as f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x = g ( f ( x
Homomorphism
flat cover of a module M over a ring is a surjective homomorphism from a flat module F to M that is in some sense minimal. Any module over a ring has
Flat_cover
Elements taken to zero by a homomorphism
set that only contains the identity if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element
Kernel_(algebra)
Homomorphisms between simple modules over the same ring are isomorphisms or zero
irreducible, V ′ {\displaystyle V'} must be zero; so f {\displaystyle f} is injective. By an identical argument we will show f {\displaystyle f} is also surjective;
Schur's_lemma
generator module is faithful, i.e. has zero annihilator. Using the Tietze extension theorem one can show that the unit interval is an injective cogenerator
Generator_(category_theory)
rings, kernels of morphisms between indecomposable injective modules, couniformly presented modules.) Right uniserial rings can also be referred to as
Serial_module
Topic in abstract algebra
global dimension ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 1. Happel (2001) classified the hereditary
Tilting_theory
Linear representation in abstract algebra
\rho :G\to GL(V)} is injective (or one-to-one). While representations of G over a field K are de facto the same as K[G]-modules (with K[G] denoting the
Faithful_representation
Equivalence relation on rings
R-Mod 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
Morita_equivalence
Mathematical operation on vector spaces
M_{1}\otimes _{R}N\to M_{2}\otimes _{R}N} is not usually injective. For example, tensoring the (injective) map given by multiplication with n, n : Z → Z with
Tensor_product
Malware that affects the Linux operating system
are able to attack by modifying anything like replacing binaries or injecting modules. This may allow the redirection of users to different content on the
Linux_malware
Ring in abstract algebra
surjective, since the image is a right ideal and contains 1. If it is not injective, then, say, a 1 y 1 = a 2 y 2 + ⋯ + a k y k {\displaystyle a_{1}y_{1}=a_{2}y_{2}+\cdots
Artinian_ring
Functor that preserves short exact sequences
k-vector spaces to itself. (Exactness follows from the above: k is an injective k-module. Alternatively, one can argue that every short exact sequence of k-vector
Exact_functor
Concept category theory (mathematics)
surjective module homomorphisms. { R → 0 } ⊥ r {\displaystyle \{R\to 0\}^{\perp r}} is the class of injective module homomorphisms. A module M {\displaystyle
Lifting_property
on one side and self-injective on one side. R is Artinian on a side and self-injective on a side. All right (or all left) R modules which are projective
Quasi-Frobenius_ring
Injective homomorphism
morphisms h : Z → X, is injective for all objects Z. Every morphism in a concrete category whose underlying function is injective is a monomorphism; in
Monomorphism
Software programming object-oriented design methodology
modules to low-level, dependency modules are reversed, thus rendering high-level modules independent of the low-level module implementation details. The principle
Dependency inversion principle
Dependency_inversion_principle
Algebraic theory
left module A that is indecomposable but not injective there is an almost-split sequence as above, which is unique up to isomorphism. The module A in
Auslander–Reiten_theory
Vector space equipped with a bilinear product
homomorphism, then one must have either that A is the zero ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication
Algebra_over_a_field
that if all the modules of a minimal injective resolution of an Artin algebra R are injective and projective, then R is self-injective. Auslander, Maurice;
Nakayama's_conjecture
Lemma in category theory about commutative diagrams
p are injective and l is surjective. Let c in C be such that n(c) = 0. t(n(c)) is then 0. By commutativity, p(h(c)) = 0. Since p is injective, h(c) =
Five_lemma
PHP web application and component framework
scanning, and code generation. Component Installer Composer plugin for injecting modules and configuration providers into application configuration. Config
Laminas
American astronaut and lunar explorer (born 1935)
astronaut, United States Air Force (USAF) officer and test pilot who, as Lunar Module pilot of Apollo 16 in 1972, became the 10th and youngest person to walk
Charles_Duke
1. The module Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } over the ring Z {\displaystyle \mathbb {Z} } has weak dimension 1, but injective dimension
Weak_dimension
Concept in algebraic geometry
− ) {\displaystyle \Gamma _{I}(-)} of an injective resolution E ∙ {\displaystyle E^{\bullet }} of the module M {\displaystyle M} . Because E ∙ {\displaystyle
Local_cohomology
Matlis module, an injective hull of k, and let Ω be the completion of its dualizing module. Then for any R-module M there is an isomorphism of modules over
Grothendieck_local_duality
Tool in algebraic topology
enough injectives; that is, for every sheaf E there is an injective sheaf I with an injection E → I. It follows that every sheaf E has an injective resolution:
Sheaf_cohomology
Objects in representation theory of Lie algebras
on infinitesimal central characters. Each homomorphism of Verma modules is injective and the dimension dim ( Hom ( W μ , W λ ) ) ≤ 1 {\displaystyle
Verma_module
1957 mathematics paper by Alexander Grothendieck
showing by this means that categories of sheaves of abelian groups admitted injective resolutions, Grothendieck went beyond the theory available in Cartan–Eilenberg
Grothendieck's_Tôhoku_paper
INJECTIVE MODULE
INJECTIVE MODULE
Surname or Lastname
Irish
Irish : reduced Anglicized form of Gaelic Ó Teimhin ‘descendant of Teimhean’, from teimhean ‘dark’, an adjective from teimhe ‘dusk’, ‘darkness’.English : probably a habitational name for someone from Tyneside in northeast England.
Boy/Male
Tamil
Valin means courage in sanskrit. adding i hence Valini would keep the meaning the same as but make it feminine as in Hindi An i at the end of a noun or adjective makes it feminine
Female
Greek
(ΑἰκατεÏίνη) Greek name of uncertain etymology, but from an early date it has been associated with the Greek adjective katharos, AIKATERINE means "pure."Â
Boy/Male
Hindu, Indian, Sanskrit
Adjective Devil
Girl/Female
Latin American Spanish
Dazzling white. Bright, glowing white. Derived from the feminine of the Latin adjective meaning...
Boy/Male
Muslim
Objective, Goal
Boy/Male
Indian
Objective, Goal
Girl/Female
African, American, Australian, British, Chinese, Danish, English, Hebrew, Indian, Irish, Japanese, Swahili
Intention; Female Champion; Aim; Objective; Goal; Purpose; Beauty; Brightness; God Gifted
Boy/Male
Arabic, Muslim
Wanted; Unknown; Objective; Goal
Surname or Lastname
North German
North German : habitational name from any of several places called Loose or Loosey.North German : from a short form of Nikolaus, German form of Nicholas.Dutch : nickname from the adjective loos ‘cunning’, ‘artful’, ‘guileful’.English : variant spelling of Loose.
Girl/Female
Indian
An Adjective to Happy as Happiest; A Type of Grass that Cuts Only Bad Person's Skin
Boy/Male
Hindu
Valin means courage in sanskrit. adding i hence Valini would keep the meaning the same as but make it feminine as in Hindi An i at the end of a noun or adjective makes it feminine
Surname or Lastname
English
English : from the informal England adjective Brummagem ‘of or relating to Birmingham’, hence a habitational name for someone from the city of Birmingham in the West Midlands.
Boy/Male
Muslim/Islamic
Objective goal
Surname or Lastname
English
English : occupational name for an officer of a court of justice, from the English vocabulary word bailiff, which is from the objective case of Old French bailis (see Bayliss).
Surname or Lastname
English
English : nickname from the adjective bony, denoting a scrawny individual with prominent bones.
Surname or Lastname
Americanized form of German, Dutch, or northern French Happe.English
Americanized form of German, Dutch, or northern French Happe.English : nickname from the adjective happy.
Surname or Lastname
English
English : habitational name from places in North and West Yorkshire named Barden, from Old English bere ‘barley’ (or the derived adjective beren) + denu ‘valley’.
Surname or Lastname
English
English : nickname for a person with a ruddy complexion, from an adjective derivative of Middle English mad(d)er ‘madder’, the dye plant (see Mader 1), here used in a transferred sense.
Girl/Female
Hindu, Indian
Queen of Horizon; Injection
INJECTIVE MODULE
INJECTIVE MODULE
Girl/Female
Latin
Lioness.
Surname or Lastname
English
English : habitational name from any of various places so called, for example in Northumberland.
Girl/Female
Hindu, Indian, Kannada, Sanskrit, Telugu, Traditional
A Bud of Champa
Boy/Male
Hindu, Indian, Jain
Lord Krishna
Female
Hebrew
(עַלִיצָה) Hebrew name ALITZA means "joy."
Boy/Male
Hindu
Pure
Girl/Female
Afghan, Arabic, Danish, Indian, Muslim, Parsi, Sanskrit
Friend
Boy/Male
Tamil
Gudakesh | கà¯à®¤à®¾à®•à¯‡à®·
Possessing thick beautiful hair
Girl/Female
Arabic, Muslim
Fortunate; Happy; Lucky; Prosperous; Gracious; Favourable; August; Feminine of Masood
Male
Norse
Usually said to be an Anglicized form of Old Norse Fenrisúlfr, but according to Sophus Bugge, author of The Home of The Eddic Poems, this name, as well as Fenrir, probably originated with Norsemen under the influence of Christianity, and was a word for "hell" and only later took on the FENRIS means "swamp."Â
INJECTIVE MODULE
INJECTIVE MODULE
INJECTIVE MODULE
INJECTIVE MODULE
INJECTIVE MODULE
n.
The objective case.
a.
Able and apt to invent; quick at contrivance; ready at expedients; as, an inventive head or genius.
a.
Not disposed to action or effort; not diligent or industrious; not busy; idle; as, an inactive officer.
a.
Facilitating induction; susceptible of being acted upon by induction; as certain substances have a great inductive capacity.
n.
A word used with a noun, or substantive, to express a quality of the thing named, or something attributed to it, or to limit or define it, or to specify or describe a thing, as distinct from something else. Thus, in phrase, "a wise ruler," wise is the adjective, expressing a property of ruler.
a.
Leading to inferences; proceeding by, derived from, or using, induction; as, inductive reasoning.
a.
Inventive.
a.
Inflective.
a.
Operating by induction; as, an inductive electrical machine.
v. t.
To make an adjective of; to form or change into an adjective.
a.
Capable of, or pertaining to, inflection; deflecting; as, the inflective quality of the air.
a.
Not active; having no power to move; that does not or can not produce results; inert; as, matter is, of itself, inactive.
n.
Added to a substantive as an attribute; of the nature of an adjunct; as, an adjective word or sentence.
n.
The act of injecting or throwing in; -- applied particularly to the forcible throwing in of a liquid, or aeriform body, by means of a syringe, pump, etc.
n.
An expression which inveighs or rails against a person; a severe or violent censure or reproach; something uttered or written, intended to cast opprobrium, censure, or reproach on another; a harsh or reproachful accusation; -- followed by against, having reference to the person or thing affected; as an invective against tyranny.
n.
Same as Objective point, under Objective, a.
a.
Not active; inert; esp., not exhibiting any action or activity on polarized light; optically neutral; -- said of isomeric forms of certain substances, in distinction from other forms which are optically active; as, racemic acid is an inactive tartaric acid.
a.
Beginning; expressing or indicating beginning; as, an inceptive proposition; an inceptive verb, which expresses the beginning of action; -- called also inchoative.
n.
A specimen prepared by injection.
n.
An inceptive word, phrase, or clause.