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Generally, something that is '''modular''' is [[construction|constructed]] so as to facilitate easy [[assembly]], flexible arrangement, and/or [[repair]].
{{for|the Wikipedia programming structure containing Lua code|Wikipedia:Module|selfref=y}}
{{Wiktionary|module|modular}}


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'''Module''', '''modular''' and '''modularity''' may refer to the concept of [[modularity]]. They may also refer to:
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In [[abstract algebra]], a <b>left ''R''-module</b> 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 written by juxtaposition, i.e. as ''rx'' for ''r'' in ''R'' and ''x'' in ''M'') such that
==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 all r,s in R, x in M: (rs)x = r(sx)
===Computer software===
  For all r,s in R, x in M: (r+s)x = rx+sx
* [[Modular programming]], a software design technique
  For all r in R, x,y in M: r(x+y) = rx+ry
** [[Java Platform Module System]]
  For all x in M: 1x = x
** [[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


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 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


If ''R'' is a [[field]], then an ''R''-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 of modules over certain rings.
==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


=== Examples ===
==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


*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'') for ''n'' < 0.
==Other uses==
*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.
* [[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]].
*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''.
* [[NTC Module]], a Russian research and development center
*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.
*[[ModulArt]], a technique used in contemporary art where a large-structure painting is made up of multiple smaller modules.
*If ''R'' is any ring and ''I'' is any left [[ideal]] in ''R'', then ''I'' is a left module over ''R''.
* [[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


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


Suppose ''M'' is an ''R''-module and ''N'' is a [[subgroup]]
{{disambiguation}}
of ''M''.  Then ''N'' is a '''submodule''' (or ''R''-submodule, to be more explicit) if, for any ''n'' in ''N'' and any ''r'' in ''R'', the product ''rn'' is in ''N'' (or ''nr'' for a right module).
 
If ''M'' and ''N'' are ''R''-modules, then a map
''f:M -> N'' is a '''homomorphism''' if, for any ''m1, m2'' in ''M''
and ''r1, r2'' in ''R'', ''f(r1m1 + r2m2) = r1f(m1) + r2f(m2)''.  This, like any [[homomorphism]] of mathematical
objects, is just a mapping which preserves the structure of the objects.
 
=== Alternative definition as representations ===
 
If ''M'' is a left ''R''-module, then the action of an element ''r'' in ''R'' is defined to be the map ''M'' &rarr; ''M'' 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<sub>'''Z'''</sub>(''M'') and forms a ring under addition and composition, and sending a ring element to its action actually defines a [[ring homomorphism]] from ''R'' to End<sub>'''Z'''</sub>(''M'').
 
Such a ring homorphism ''R'' &rarr; End<sub>'''Z'''</sub>(''M'') is called a ''representation'' of ''R'' over the abelian group ''M''; an alternative and equivalent way to defining left ''R''-modules is to say that a left ''R''-module is an abelian group ''M'' together with a representation of ''R'' over it.
 
A representation is called ''faithful'' if and only if the map ''R'' &rarr; End<sub>'''Z'''</sub>(''M'') is [[injective|one-to-one]]. Every abelian group is a module over the [[integer|integers]], and is either faithful under them or some [[modular arithmetic]].

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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