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In [[abstract algebra]], a '''left ''R''-module''' consists of some commutative [[Mathematical Group|group]] (M,+) together with a [[Mathematical ring|ring]] of scalars (''R'',+,*) and an operation ''R'' x ''M'' <tt>-></tt> ''M'' (scalar multiplication, usually just denoted *) such that
{{for|the Wikipedia programming structure containing Lua code|Wikipedia:Module|selfref=y}}
{{Wiktionary|module|modular}}


'''Module''', '''modular''' and '''modularity''' may refer to the concept of [[modularity]]. They may also refer to:
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==Computer science and engineering==
* [[Modular design]], the engineering discipline of designing complex devices using separately designed sub-components
* [[Modular function deployment]], a method in systems engineering and product development
* [[Gear#Standard pitches and the module system|Module]], a measure of a gear's pitch
* [[Ontology modularization]], a methodological principle in ontology engineering


  For r,s in R, x in M, (rs)x = r(sx)
===Computer software===
* [[Modular programming]], a software design technique
** [[Java Platform Module System]]
** [[Modules (C++)]]
* [[Module SQL]]
* [[Loadable kernel module]], an object file that contains code to extend the running kernel
* [[Modules Environment|Environment Modules]], a software tool designed to help users manage their Unix or Linux shell environment
* [[Modula-2]] or [[Modula-3]], programming languages which stress the use of modules


  For r,s in R, x in M, (r+s)x = rx+sx
===Computer hardware===
* [[Computer module]], an early packaging technique that combined several electronic components to produce a single logic element
* [[Memory module]], a physical "stick" of RAM, an essential piece of computer hardware
* [[Multi-chip module]], a modern technique that combines several complex computer chips into a single larger unit


  For r in R, x,y in M, r(x+y) = rx+ry
==Science and mathematics==
* [[Module (mathematics)]] over a ring, a generalization of vector spaces
* [[G-module|''G''-module]] over a group ''G'', in mathematics
* [[Modular lattice]] a kind of partially ordered set
* [[Modularity theorem]] (formerly Taniyama–Shimura conjecture), a connection between elliptic curves and modular forms
* Module, in connection with [[modular decomposition]] of a graph, a kind of generalisation of graph components
* [[Modularity (networks)]], a benefit function that measures the quality of a division of a [[Complex network]] into communities
* Protein module or [[protein domain]], a section of a protein with its own distinct conformation, often conserved in evolution
* A [[Cis-regulatory module|''cis''-regulatory module]], a stretch of DNA containing a number of genes that share joint regulation by the same transcription factors


  For x in M, 1x = x
==Music==
* [[Module (musician)]], the solo project of New Zealand-based musician/producer Jeramiah Ross
* [[Module file]], a family of music file formats
* [[Modular Recordings]], a record label
* [[Modular synthesizer]], a type of electronic musical instrument
* [[Sound module]], electronic musical instrument without a human-playable interface


==Other uses==
* [[Modular building]]: prefabricated building that consists of repeated sections called modules, used as house or other, some of them open source, in this case, [[open source hardware]].
* [[NTC Module]], a Russian research and development center
*[[ModulArt]], a technique used in contemporary art where a large-structure painting is made up of multiple smaller modules.
* [[Ford Modular engine]], Ford's line of OHC V8 and V10 motors
* [[Volvo Modular engine]]
* [[Game module]] or expansion, an add-on publication for a role-playing game
** [[Adventure (Dungeons & Dragons)]], formerly referred to as a ''module''
* [[Vitruvian module]], an architectural measure
* A class, [[course (education)|course]], or unit of education covering a single topic
* [[Modular AI]], an American company


==See also==
* [[Modulus (disambiguation)]]
* [[Atomicity (disambiguation)]]
* [[Modul University Vienna]]
* [[Modulon]]


A right R-module is defined similarly, only the ring acts on the right.  The two are easily interchangeable.
{{disambiguation}}
 
 
 
The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an [[endomorphism]] of M.  The set of all endomorphisms of M is denoted End(M) and forms a ring under addition and composition, so the above actually defines a [[homomorphism]] from R into End(M).
 
 
 
This is called a representation of R over M, and is called faithful if and only if the map is one-to-one.  M can be expressed as an R-module if and only if R has some representation over it.  In particular, every commutative group is a module over the [[integer]]s, and is either faithful under them or some modular arithmetic.
 
 
 
Another thing to note is that End(M), treated as a group, is also a module over R in a natural way.  When R is a [[field]], this constitutes an associative algebra.  Modules over fields are called [[vector space]]s.
 
 

Latest revision as of 23:09, 27 September 2025

Template:For Template:Wiktionary

Module, modular and modularity may refer to the concept of modularity. They may also refer to: Template:TOC right

Computer science and engineering

Computer software

Computer hardware

  • Computer module, an early packaging technique that combined several electronic components to produce a single logic element
  • Memory module, a physical "stick" of RAM, an essential piece of computer hardware
  • Multi-chip module, a modern technique that combines several complex computer chips into a single larger unit

Science and mathematics

Music

Other uses

See also

Template:Disambiguation

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