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Type of module over a ring
simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M
Simple_module
Generalization of vector spaces from fields to rings
R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that
Module_(mathematics)
Direct sum of irreducible modules
area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from
Semisimple_module
Mathematical property
trivial module 0 is semi-simple, but not simple). For an R-module M, M is semi-simple if and only if it is the direct sum of simple modules (the trivial
Semi-simplicity
Homomorphisms between simple modules over the same ring are isomorphisms or zero
and the vector space M = N is a simple module of R. Then Schur's lemma says that the endomorphism ring of the module M is a division algebra over k. If
Schur's_lemma
Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" N < M, while
Indecomposable_module
Studies linear representations of finite groups over fields of positive characteristic
the number of simple modules k(G) is equal to the number of conjugacy classes of G. In the modular case, the number l(G) of simple modules is equal to the
Modular_representation_theory
Decomposition of an algebraic structure
structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that
Composition_series
In algebra, integer associated to a module
intersection theory. The zero module is the only one with length 0. Modules with length 1 are precisely the simple modules. The length of the cyclic group
Length_of_a_module
Abstract algebra concept
example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can
Decomposition_of_a_module
Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject. See also: Glossary of linear
Glossary_of_module_theory
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
Ideal of a ring contained in no other ideal except the ring itself
ideal B with A ⊆ B, either B = A or B = R. The quotient module R/A is a simple right R-module. Maximal right/left/two-sided ideals are the dual notion
Maximal_ideal
Index of articles associated with the same name
subgroup. A ring is called a simple ring if it does not contain a nontrivial two sided ideal. A module is called a simple module if it does not contain a
Simple_(abstract_algebra)
as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simple modules, projective
Principal indecomposable module
Principal_indecomposable_module
finite-dimensional module over g {\displaystyle {\mathfrak {g}}} is semisimple as a module (i.e., a direct sum of simple modules.) Weyl's theorem implies
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
Mathematical object in abstract algebra
abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers
Injective_module
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
Structure in Ring Theory (Mathematics)
those elements in R {\displaystyle R} that annihilate all simple right R {\displaystyle R} -modules. It happens that substituting "left" in place of "right"
Jacobson_radical
Algebraic structure also called skew field
arises in this fashion from some simple module. Much of linear algebra may be formulated, and remains correct, for modules over a division ring D instead
Division_ring
Type of group and algebra representation
analogous to an irreducible representation in the resulting theory is a simple module.[citation needed] Let ρ {\displaystyle \rho } be a representation i
Irreducible_representation
Algebra in statistical mechanics
mod 2 {\displaystyle \ell \equiv n{\bmod {2}}} . The dimension of a simple module is written in terms of binomial coefficients as dim ( W ℓ ) = ( n
Temperley–Lieb_algebra
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
Mathematical theorem
ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R. The theorem can be
Jacobson_density_theorem
Type of ring in non-commutative algebra
non-trivial two-sided ideals is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring
Simple_ring
Algebraic structure
A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible)
Noncommutative_ring
Algebraic structure with addition and multiplication
A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left module (or right module) over itself.
Ring_(mathematics)
Module over the non-commutative Dieudonné ring
The category of such modules is semisimple, so every module is a direct sum of simple modules. The simple modules are the modules E s / r {\displaystyle
Dieudonné_module
-module. A submodule N {\displaystyle N} of M {\displaystyle M} is called maximal or cosimple if the quotient M / N {\displaystyle M/N} is a simple module
Radical_of_a_module
Statement in abstract algebra
Rn by a particularly simple submodule, and this is the structure theorem. The structure theorem for finitely generated modules over a principal ideal
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Direct sum of simple Lie algebras
simple g {\displaystyle {\mathfrak {g}}} -module V μ {\displaystyle V^{\mu }} having μ {\displaystyle \mu } as its highest weight and (2) two simple modules
Semisimple_Lie_algebra
Writing Lie algebra sets as matrices
{g}}W} is simple (resp. absolutely simple), then W is simple (resp. absolutely simple). Here, a g {\displaystyle {\mathfrak {g}}} -module V is absolutely
Lie_algebra_representation
Submodule of a mathematical ring
ideal is finitely generated as a module. Primitive ideal: A left primitive ideal is the annihilator of a simple left module. Irreducible ideal: An ideal is
Ideal_(ring_theory)
Direct summand of a free module (mathematics)
projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various
Projective_module
R-module N is cyclic if N = yR for some y ∈ N. 2Z as a Z-module is a cyclic module. In fact, every cyclic group is a cyclic Z-module. Every simple R-module
Cyclic_module
A particular algebraic structure
such that every simple R-module is injective. The following three conditions are equivalent: Every simple left (respectively right) R-module is injective
V-ring_(ring_theory)
Annihilator of a simple module
theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive
Primitive_ideal
Algebra over a field with only invertible elements and zero
Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F;
Division_algebra
Maps whose domain and codomain are acted on by the same group, and the map commutes
hold when the image of K[G] is a simple algebra, with centre K (by what is called Schur's lemma: see simple module). As a consequence, in important cases
Equivariant_map
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
Index of articles associated with the same name
proper subrepresentations. Similarly, an irreducible module is another name for a simple module. Absolutely irreducible is a term applied to mean irreducible
Irreducibility_(mathematics)
Concept in Lie algebra representation theory
isomorphism) simple highest-weight g {\displaystyle {\mathfrak {g}}} -module with highest weight λ, which is denoted L(λ), but this module is infinite
Weight (representation theory)
Weight_(representation_theory)
Deep learning software
can be joined using module composites, like Sequential, Parallel and Concat to create complex task-tailored graphs. Simpler modules like Linear, Tanh and
Torch_(machine_learning)
Notion in abstract algebra
algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of
Injective_hull
Branch of mathematics that studies algebraic structures
(ring theory) Simple module, Semisimple module Indecomposable module Artinian module, Noetherian module Homological types: Projective module Projective cover
List of abstract algebra topics
List_of_abstract_algebra_topics
URL parameter used for tracking
Urchin Tracking Module (UTM) parameters are five variants of URL parameters used by marketers to track the effectiveness of online marketing campaigns
UTM_parameters
Computer architecture for educational purposes
(7400-series) capable of running simple programs such as computing the Fibonacci sequence. Eater's design consists of the following modules: An adjustable-speed (upper
Simple-As-Possible_computer
Graphics and Multimedia Library written in C++
a graphics module for simple hardware acceleration of 2D computer graphics which includes text rendering using FreeType, an audio module that uses OpenAL
Simple and Fast Multimedia Library
Simple_and_Fast_Multimedia_Library
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++)
Term in abstract algebra
{\displaystyle L(\lambda )} are absolute simple A {\displaystyle A} -modules and every simple A {\displaystyle A} -module is one of these. These theorems appear
Cellular_algebra
Concerns the decomposition of representations of a finite group into irreducible pieces
representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context
Maschke's_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
Software platform
third-party modules. The Ignition platform has three main components: the Ignition Gateway, the Designer, and the runtime clients. Independent modules provide
Ignition_SCADA
In mathematics, element that equals its square
A right irreducible idempotent is an idempotent a for which aR is a simple module. By Schur's lemma, EndR(aR) = aRa is a division ring, and hence is a
Idempotent_(ring_theory)
Instructions a computer can execute
object-oriented module is composed of two files. The definitions file is called the header file. Here is a C++ header file for the GRADE class in a simple school
Computer_program
This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Rings such as the ring
Semiprimitive_ring
European Declaration of conformity mark
(Module) to be implemented: Module A – Internal production control. Module A1 – Internal production control plus supervised product testing Module A2
CE_marking
Ring that is also a vector space or a module
and scalar multiplication operations together give A the structure of a module or vector space over K. In this article we will also use the term K-algebra
Associative_algebra
uniserial modules are serial modules. It has been verified that Jacobson's conjecture holds in Noetherian serial rings. Any simple module is trivially
Serial_module
In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field
Quillen's_lemma
Classification of semi-simple rings and algebras
the right R {\displaystyle R} -module R R {\displaystyle R_{R}} is isomorphic to a finite direct sum of simple modules (which are the same as minimal
Wedderburn–Artin_theorem
Index of articles associated with the same name
radical of a module. In set notation, s o c ( M ) = ∑ N is a simple submodule of M N . {\displaystyle \mathrm {soc} (M)=\sum _{N{\text{ is a simple submodule
Socle_(mathematics)
Algebraic structure with "nice" duality properties
whose simple modules M have the same dimension as their A-duals, HomA(M,A). Amongst these algebras, the A-duals of simple modules are always simple. A finite-dimensional
Frobenius_algebra
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
Ring in abstract algebra
MR 1314422 Bourbaki, Nicolas (2012). Algèbre. Chapitre 8, Modules et anneaux semi-simples. Heidelberg: Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-35315-7
Artinian_ring
Lightweight Linux distribution
with all files contained in an initramfs built into the kernel. It has simple module loading management but fewer drivers are included. It is used for experimental
Puppy_Linux
Microcontroller peripheral interface
may also refer to modules compatible with the Pmod interface. Pmods are available from simple push buttons to more complex modules with network interfaces
Pmod_Interface
Integral polynomial
dimensional representations of semisimple Lie algebras, given by Verma modules and simple modules. This analogy, and the work of Jens Carsten Jantzen and Anthony
Kazhdan–Lusztig_polynomial
Set with associative invertible operation
of finite simple groups, for example. See Aschbacher 2004. See, for example, Schur's Lemma for the impact of a group action on simple modules. A more involved
Group_(mathematics)
Color used in designing web pages
The working draft for the level 4 color module combines the Basic and Extended sections together in a simple "Named Colors" section. CSS 2, SVG and CSS
Web_colors
Commutative group (mathematics)
finite simple abelian groups are exactly the cyclic groups of prime order. The concepts of abelian group and Z {\displaystyle \mathbb {Z} } -module agree
Abelian_group
System-on-a-chip microcontroller model with Wi-Fi
ESP-01 module, made by a third-party manufacturer Ai-Thinker. This small module allows microcontrollers to connect to a Wi-Fi network and make simple TCP/IP
ESP8266
Internet forum software
ca. 28 October 2018. "Vanilla". homebrewforums.net. "How to Build a Simple Module and Integrate it as a Plugin". homebrewforums.net. "Releases · vanilla/vanilla"
Vanilla_Forums
ideal of R with {0} ⊆ K ⊆ N, then either K = {0} or K = N. N is a simple right R-module. Minimal ideals are the dual notion to maximal ideals. Many standard
Minimal_ideal
together. T-TRAK modules are dioramas with sectional track, specifically Kato Unitrack, that snap together to create layouts from a simple circle to large
T-Trak
Branch of algebra
structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division rings, universal enveloping
Ring_theory
Sixth and most recent crewed Moon landing
the Moon. Commander Gene Cernan and Lunar Module Pilot Harrison Schmitt walked on the Moon, while Command Module Pilot Ronald Evans orbited above. Schmitt
Apollo_17
Synthesizer composed of separate modules
synthesizer modules that represent different functions. The modules can be connected together by the user to create a patch. The outputs from the modules may
Modular_synthesizer
Interpreted server-side scripting language
2018-02-09. Retrieved 2018-02-09. "Module ngx_http_ssi_module". nginx documentation. Retrieved 16 November 2021. "Apache Module mod_include". Apache HTTP Server
Server_Side_Includes
Module over a sheaf of differential operators
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of
D-module
the double centralizer theorem states that simple right modules for right Artinian rings are balanced modules. They are full matrix rings over polynomial
Double_centralizer_theorem
common functions are in libraries which are loaded on demand. Module system - A simple module system, API compatible to STklos. Object system - CLOS-like
Gauche (Scheme implementation)
Gauche_(Scheme_implementation)
Modular space station in low Earth orbit
plans for the ISS include the addition of at least one module, the Payload Power Thermal Module by Axiom Space, forming the commercial segment of the station
International_Space_Station
Element in a ring whose some power is 0
characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of a ring R {\displaystyle
Nilpotent
Computer network management and monitoring protocol
Simple Network Management Protocol (SNMP) is an Internet Standard protocol for collecting and organizing information about managed devices on IP networks
Simple Network Management Protocol
Simple_Network_Management_Protocol
mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra C is a central simple algebra over some field
Clifford_module
Type of standardized secure cryptoprocessors
A Trusted Platform Module (TPM) is a secure cryptoprocessor that implements the ISO/IEC 11889 standard. Common uses are verifying that the boot process
Trusted_Platform_Module
ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and
Primitive_ring
Chinese modular space station (since 2021)
food, instead of simple reheating. Chinese astronauts cooked barbecue chicken wings and grilled steak with it. The station's core module, Tianhe, provides
Tiangong_space_station
Modular communications interface
Form-factor Pluggable (SFP) is a compact, hot-pluggable network interface module format used for both telecommunication and data communications applications
Small_Form-factor_Pluggable
Family of web feed formats
RSS (RDF Site Summary or Really Simple Syndication) is a web feed that allows users and applications to access updates to websites in a standardized, computer-readable
RSS
Theorem representing a solvable Lie algebra
is simple; thus, is one-dimensional. The induction now finishes the proof. The statement says in particular that a finite-dimensional simple module over
Lie's_theorem
Guidance and navigation computer used in Apollo spacecraft
program that was installed on board each Apollo command module (CM) and Apollo Lunar Module (LM). The AGC provided computation and electronic interfaces
Apollo_Guidance_Computer
Optical transceiver that uses coherent modulation
have been employed in coherent optical modules. Some coherent optical modules can fall back to older, simpler modulation techniques such as on-off keying
Coherent_optical_module
Degree of interdependence between software modules
between software modules, a measure of how closely connected two routines or modules are, and the strength of the relationships between modules. Coupling is
Coupling (computer programming)
Coupling_(computer_programming)
Style sheet language
features, CSS 3 is divided into several separate documents called "modules". Each module adds new capabilities or extends features defined in CSS 2, preserving
CSS
1961–1972 American crewed lunar exploration program
Lunar Module (LM) on July 20, 1969, and walked on the lunar surface, while Michael Collins remained in lunar orbit in the command and service module (CSM)
Apollo_program
Messaging protocol for web services
SOAP (originally an acronym for Simple Object Access Protocol) is a messaging protocol specification for exchanging structured information in the implementation
SOAP
G-submodules of k [ X ] {\displaystyle k[X]} that are isomorphic to the simple module V λ {\displaystyle V^{\lambda }} ; it is called the λ {\displaystyle
Representation on coordinate rings
Representation_on_coordinate_rings
Rings admitting weak inverses
16500 Michler, G.O.; Villamayor, O.E. (April 1973), "On rings whose simple modules are injective", Journal of Algebra, 25 (1): 185–201, doi:10.1016/0021-8693(73)90088-4
Von_Neumann_regular_ring
SIMPLE MODULE
SIMPLE MODULE
Surname or Lastname
English
English : variant spelling of Kimball.English : habitational name from Great or Little Kimble in Buckinghamshire, named in Old English as ‘the royal bell’ (cynebelle), referring to the shape of a local hill.Americanized spelling of German Gimbel (see Gimble) or Kimbel.
Girl/Female
Indian
A small indication one that forms in the cheeks when one smiles
Female
French
 Feminine form of French Simon, SIMONE means "hearkening." Compare with other forms of Simone.
Surname or Lastname
English (Kent)
English (Kent) : origin uncertain; perhaps a variant of the habitational name Wimbley, or a variant of Wimple, a metonymic occupational name for a maker of wimples, from Middle English wimple (Old English wimpel ‘veil’).
Boy/Male
English
Temple-town. This surname refers to medieval priories and settlements of the military religious...
Female
Icelandic
 Feminine form of Icelandic SÃmon, SIMONE means "hearkening." Compare with other forms of Simone.
Male
Italian
Italian form of Hebrew Shimown, SIMONE means "hearkening."
Boy/Male
Shakespearean
The Merry Wives of Windsor' Servant to Slender.
Surname or Lastname
English (mainly Nottinghamshire)
English (mainly Nottinghamshire) : unexplained; probably a variant of Sample.
Female
Finnish
 Feminine form of Finnish Simo, SIMONE means "hearkening." Compare with another form of Simone.
Girl/Female
Hindu, Indian, Kannada
Loved One
Girl/Female
Indian, Telugu
Simple Looking; Good Smile
Female
Scandinavian
 Scandinavian feminine form of Greek Symeon, SIMONE means "hearkening." Compare with other forms of Simone.
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from any of various places in Normandy called Saint-Paul or Saint-Pol, from the dedication of their churches to St. Paul (see Paul).
Girl/Female
American, Assamese, British, Celebrity, English, Gujarati, Hindu, Indian, Kannada, Malayalam, Sindhi, Telugu
A Small; Natural Hollow on the Surface of the Body; Happy; Dimples
Boy/Male
Hindu, Indian
Soft; Gentle Spirit with a Profound Spiritual Nature
Boy/Male
Australian, British, English
From the Temple Settlement
Surname or Lastname
English
English : from Middle English stapel ‘post’, hence a topographic name for someone who lived near a boundary post, or a habitational name from some place named with this word (Old English stapel), as for example Staple in Kent or Staple Fitzpaine in Somerset.Americanized spelling of German Stapel.
Boy/Male
Indian
Chick Style
Girl/Female
Indian
Beauty
SIMPLE MODULE
SIMPLE MODULE
Surname or Lastname
German
German : topographic name for someone who lived by a stream, Middle High German bach ‘stream’. This surname is established throughout central Europe and in Scandinavia, not just in Germany.Jewish (Ashkenazic) : ornamental name from German Bach ‘stream’, ‘creek’.English : topographic name for someone who lived by a stream, Middle English bache.Welsh : distinguishing epithet from Welsh bach ‘little’, ‘small’.Norwegian : Americanized spelling of the topographic name Bakk(e) ‘hillside’ (see Bakke).Polish, Czech, and Slovak : from the personal name Bach, a pet form of Bartomolaeus (Polish Bartłomiej, Czech Bartoloměj, Slovak Bartolomej (see Bartholomew) or possibly in some cases of Baltazar or Sebastian).
Surname or Lastname
English
English : see Sanborn.Swedish : ornamental name from sand ‘sand’ + born ‘well’.
Girl/Female
Arabic, Swahili
Woman; Life
Girl/Female
Hindu
Male
French
Variant form of Norman French Emaurri, AMAURY means "work-power."
Girl/Female
Irish
The feminine of the name Aidan meaning “little fire.â€
Female
English
English name derived from the country name, INDIA means "land of the Indus (river)."Â
Boy/Male
Indian, Punjabi, Sikh
Support of the Supreme God
Boy/Male
Indian
Luck
Male
Egyptian
, a son of King Aahmes I.
SIMPLE MODULE
SIMPLE MODULE
SIMPLE MODULE
SIMPLE MODULE
SIMPLE MODULE
v. t.
To cause to appear as if laid in folds or plaits; to cause to ripple or undulate; as, the wind wimples the surface of water.
v. t. & i.
To rumple; to wrinkle.
v. i.
To gather simples, or medicinal plants.
a.
Artless; guileless; simple-hearted; undesigning; unsuspecting; devoid of duplicity.
pl.
of Simile
a.
Single; not complex; not infolded or entangled; uncombined; not compounded; not blended with something else; not complicated; as, a simple substance; a simple idea; a simple sound; a simple machine; a simple problem; simple tasks.
a.
Full of dimples, or small depressions; dimpled; as, the dimply pool.
a.
Simple; not wise; weak; silly.
a.
Plain; unadorned; as, simple dress.
a.
Direct; clear; intelligible; not abstruse or enigmatical; as, a simple statement; simple language.
a.
A medicinal plant; -- so called because each vegetable was supposed to possess its particular virtue, and therefore to constitute a simple remedy.
v. t.
To take or to test a sample or samples of; as, to sample sugar, teas, wools, cloths.
a.
Not luxurious; without much variety; plain; as, a simple diet; a simple way of living.
imp. & p. p.
of Dimple
imp. & p. p.
of Rimple
a.
Without subdivisions; entire; as, a simple stem; a simple leaf.
n.
Fig.: A swelling or protuberance like a pimple.
a.
Consisting of a single individual or zooid; as, a simple ascidian; -- opposed to compound.
n.
One who makes up samples for inspection; one who examines samples, or by samples; as, a wool sampler.
a.
Not capable of being decomposed into anything more simple or ultimate by any means at present known; elementary; thus, atoms are regarded as simple bodies. Cf. Ultimate, a.