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In [[abstract algebra]], a '''left ''R''-module''' consists of some abelian [[Mathematical Group|group]] (M,+) together with a [[ring (algebra)|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}}


  For r,s in R, x in M, (rs)x = r(sx)
'''Module''', '''modular''' and '''modularity''' may refer to the concept of [[modularity]]. They may also refer to:
  For r,s in R, x in M, (r+s)x = rx+sx
<!--PLEASE RESPECT ALPHABETICAL ORDER-->
  For r in R, x,y in M, r(x+y) = rx+ry
{{TOC right}}
  For x in M, 1x = x


A right ''R''-module is defined similarly, only the ring acts on the right. The two notions are identical if the ring ''R'' is commutative.
==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


If ''R'' is a [[field]], then a module is also called a [[vector space]]. Modules are thus generalizations of vector spaces, and much of the theory of modules consists of recovering desirable properties of vector spaces in the realm or modules over certain rings.
===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


=== Examples ===
===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


*Every abelian group ''M'' is a module over the ring of [[integer|integers]] '''Z''' if we define ''nx'' = ''x'' + ''x'' + ... + ''x'' (''n'' summands) for ''n'' &ge; 0, and (-''n'')''x'' = -(''nx'').
==Science and mathematics==
*If ''R'' is any ring and ''n'' a [[natural number]], then the [[cartesian product]] ''R''<sup>''n''</sup> is a module over ''R'' if we use the component-wise operations.
* [[Module (mathematics)]] over a ring, a generalization of vector spaces
*If ''M'' is a smooth [[manifold]], then the smooth functions from ''M'' to the [[real number|real numbers]] form a ring ''R''. The set of all vector fields defined on ''M'' form a module over ''R'', and so do the tensor fields and the differential forms on ''M''.
* [[G-module|''G''-module]] over a group ''G'', in mathematics
*The square ''n''-by-''n'' [[matrix|matrices]] with real entries form a ring ''R'', and the [[Euclidean space]] '''R'''<sup>''n''</sup> is a left module over this ring if we define the module operation via matrix multiplication.
* [[Modular lattice]] a kind of partially ordered set
*If ''R'' is any ring and ''I'' is any left [[ideal]] in ''R'', then ''I'' is a left module over ''R''.
* [[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


=== Submodules and homomorphisms ===
==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


:''Still missing.''
==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


=== Alternative definition as representations ===
==See also==
 
* [[Modulus (disambiguation)]]
The action of an element ''r'' in ''R'' is defined to be the map that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a [[group homomorphism|group endomorphism]] of ''M''.  The set of all group endomorphisms of ''M'' is denoted End(''M'') and forms a ring under addition and composition, and the actions of ring elements actually define a [[ring homomorphism]] from ''R'' to End(''M'').
* [[Atomicity (disambiguation)]]
 
* [[Modul University Vienna]]
Such a homorphism 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 abelian group is a module over the [[integer|integers]], and is either faithful under them or some modular arithmetic.
* [[Modulon]]


{{disambiguation}}

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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