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[[pl:Modu%C5%82 (matematyka)]]
{{for|the Wikipedia programming structure containing Lua code|Wikipedia:Module|selfref=y}}
{{Wiktionary|module|modular}}


Generally, something that is '''modular''' is [[construction|constructed]] so as to facilitate easy [[assembly]], flexible arrangement, and/or [[repair]].
'''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


In [[abstract algebra]], a <b>left ''R''-module</b> consists of an abelian [[Mathematical Group|group]] (''M'', +) together with a [[ring (algebra)|ring]] of scalars (''R'',+,*) and an operation ''R'' &times; ''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 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 all ''r'',''s'' in ''R'', ''x'',''y'' in ''M'', we have
===Computer hardware===
# (''rs'')''x'' = ''r''(''sx'')
* [[Computer module]], an early packaging technique that combined several electronic components to produce a single logic element
# (''r''+''s'')''x'' = ''rx''+''sx''
* [[Memory module]], a physical "stick" of RAM, an essential piece of computer hardware
# ''r''(''x''+''y'') = ''rx''+''ry''
* [[Multi-chip module]], a modern technique that combines several complex computer chips into a single larger unit
# 1''x'' = ''x''


Usually, we simply write "a left ''R''-module ''M''" or <sub>''R''</sub>''M''.
==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


Some authors omit condition 4 for the general defintition of left modules, and call the above defined structures "unital left modules". In this encyclopedia however, all modules are assumed to be unital.
==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


A <b>right ''R''-module</b> ''M'' or ''M''<sub>''R''</sub> is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form ''M'' &times; ''R'' <tt>-></tt> ''M'', and the above three axioms are written with scalars ''r'' and ''s'' on the right of ''x'' and ''y''. If ''R'' is [[commutative ring|commutative]], then the left ''R''-module is the same as the right ''R''-module and is simply called an ''R''-module.
==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


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.  However, in general, an ''R''-module may not have a [[basis (linear algebra)|basis]].
==See also==
* [[Modulus (disambiguation)]]
* [[Atomicity (disambiguation)]]
* [[Modul University Vienna]]
* [[Modulon]]


=== Examples ===
{{disambiguation}}
 
*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'' > 0, 0''x'' = 0, and (-''n'')''x'' = -(''nx'') for ''n'' < 0.
*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.
*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''.
*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.
*If ''R'' is any ring and ''I'' is any left [[ring ideal|ideal]] in ''R'', then ''I'' is a left module over ''R''.
 
=== Submodules and homomorphisms ===
 
Suppose ''M'' is an ''R''-module and ''N'' is a [[subgroup]]
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 left ''R''-modules, then a [[function|map]]
''f'' : ''M'' <tt>-></tt> ''N'' is a '''homomorphism of <i>R</i>-modules''' if, for any ''m, n'' in ''M''
and ''r, s'' in ''R'', ''f''(''rm'' + ''sn'') = ''rf''(''m'') + ''sf''(''n'').  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 the abelian group (''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 ''r'' of ''R'' 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 of 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]]. In terms of modules, this means that if ''r'' is an element of ''R'' such that ''rx''=0 for all ''x'' in ''M'', then ''r''=0. Every abelian group is a faithful module over the [[integer|integers]] or over some [[modular arithmetic]] '''Z'''/''n'''''Z'''.

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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