en>Tom.Reding: Gen fixes (page/s, endash, , et al., unicodify, and/or concising wikilinks, etc.), & ref cleanup using AWB

2015-01-05T17:15:19Z

Gen fixes (page/s, endash, &nbsp, et al., unicodify, and/or concising wikilinks, etc.), & ref cleanup using AWB

207.255.205.112: /* Particle accelerators */

2014-02-03T01:48:25Z

Particle accelerators

← Older revision		Revision as of 03:48, 3 February 2014
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	~~Hi there~~. ~~My title~~ is ~~Sophia Meagher although it~~ is ~~not~~ the ~~title~~ on ~~my beginning certification~~. ~~What me~~ and ~~my family members adore~~ is to ~~climb but I~~'~~m thinking on beginning some thing new~~. ~~Alaska~~ is the only ~~place I~~'~~ve been residing~~ in ~~but now I~~'~~m contemplating other options~~. ~~My working day occupation is an information officer but I~~'~~ve currently applied~~ for ~~an additional one~~.<br><br>~~Here~~ is ~~my blog post - best psychic~~ ([~~http~~://~~linkingtear~~.~~com~~/~~profile~~-~~17739~~/~~info~~/ ~~http~~://~~linkingtear~~.~~com~~/])		In [[mathematics]], a '''free module''' is a [[free object]] in a category of [[module (mathematics)\|module]]s. Given a set <math>S</math>, a free module on <math>S</math> is a free module with [[basis]] <math>S</math>.

			Every [[vector space]] is free,<ref>Keown (1975), {{Google books quote\|id=hC9iTw8DO7gC\|page=24\|text=Every vector space is free\|p. 24}}</ref> and the free vector space on a [[Set (mathematics)\|set]] is a special case of a free module on a set.

			==Definition==
			A free module is a [[module (mathematics)\|module]] with a [[Basis (linear algebra)\|basis]]:<ref>Hazewinkel (1989), {{Google books quote\|id=s9F71NJxwzoC\|page=110\|text=A free module is a module with a basis\|p. 110}}</ref> a [[linearly independent]] [[finitely generated module\|generating set]].

			For an <math>R</math>-module <math>M</math>, the set <math>E\subseteq M</math> is a basis for <math>M</math> if:
			# <math>E</math> is a generating set for <math>M</math>; that is to say, every element of <math>M</math> is a finite sum of elements of <math>E</math> multiplied by coefficients in <math>R</math>;
			# <math>E</math> is linearly independent, that is, if <math>r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M</math> for <math>e_1, e_2, \ldots , e_n</math> distinct elements of <math>E</math>, then <math>r_1 = r_2 = \cdots = r_n = 0_R</math> (where <math>0_M</math> is the zero element of <math>M</math> and <math>0_R</math> is the zero element of <math>R</math>).

			If <math>R</math> has [[invariant basis number]], then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the '''rank''' of the free module <math>M</math>, and <math>M</math> is said to be ''free of rank n'', or simply ''free of finite rank'' if the cardinality is finite.

			Note that an immediate [[corollary]] of (2) is that the coefficients in (1) are unique for each <math>x</math>.

			The definition of an infinite free basis is similar, except that <math>E</math> will have infinitely many elements. However the sum must still be finite, and thus for any particular <math>x</math> only finitely many of the elements of <math>E</math> are involved.

			In the case of an infinite basis, the rank of <math>M</math> is the [[cardinality]] of <math>E</math>.

			==Construction==

			Given a set <math>E</math>, we can construct a free <math>R</math>-module over <math>E</math>. The module is simply the [[direct sum of modules\|direct sum]] of [[Cardinality\|<math>\|E\|</math>]] copies of <math>R</math>, often denoted <math>R^{(E)}</math>. We give a concrete realization of this direct sum, denoted by <math>C(E)</math>, as follows:
			* '''Carrier:''' <math>C(E)</math> contains the functions <math>f:E\to R</math> such that <math>f(x)=0</math> for [[cofinite\|cofinitely many]] (all but finitely many) <math>x\in E</math>.

			* '''Addition:''' for two elements <math>f,g\in C(E)</math>, we define <math>f+g\in C(E)</math> by <math>(f+g)(x) = f(x) + g(x), \forall x\in E</math>.

			* '''Inverse:''' for <math>f\in C(E)</math>, we define <math>(-f)\in C(E)</math> by <math>(-f)(x) = -(f(x)), \forall x\in E</math>.

			* '''Scalar multiplication:''' for <math>\alpha\in R, f\in C(E)</math>, we define <math>\alpha f\in C(E)</math> by <math>(\alpha f)(x) = \alpha (f(x)), \forall x\in E</math>.

			A basis for <math>C(E)</math> is given by the set <math>\{\delta_a : a\in E\}</math> where
			:<math> \delta_a(x) = \begin{cases} 1, \quad\mbox{if } x=a; \\ 0, \quad\mbox{if } x\neq a \end{cases} </math>
			(a variant of the [[Kronecker delta]] and a particular case of the [[indicator function]], for the set <math>\{a\}</math>).

			Define the mapping <math>\iota : E\to C(E)</math> by <math>\iota(a) = \delta_a</math>. This mapping gives a bijection between <math>E</math> and the basis vectors <math>\{\delta_a\}_{a\in E}</math>. We can thus identify these sets. Thus <math>E</math> may be considered as a linearly independent basis for <math>C(E)</math>.

			==Universal property==

			The mapping <math>\iota : E\to C(E)</math> defined above is [[universal property\|universal]] in the following sense. If there is an arbitrary <math>R</math>-module <math>M</math> and an arbitrary mapping <math>\varphi : E\to M</math>, then there exists a unique [[module homomorphism]] <math>\psi : C(E)\to M</math> such that <math>\varphi = \psi\circ\iota</math>.

			==Generalisations==

			Many statements about free modules, which are wrong for general modules over rings, are still true for certain generalisations of free modules. [[Projective modules]] are direct summands of free modules, so one can choose an injection in a free module and use the basis of this one to prove something for the projective module. Even weaker generalisations are [[flat module]]s, which still have the property that tensoring with them preserves exact sequences, and [[Torsion (algebra)\|torsion-free modules]]. If the ring has special properties, this hierarchy may collapse, i.e. for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well.

			:[[File:Module properties in commutative algebra.svg\|Module properties in commutative algebra]]

			See [[local ring]], [[perfect ring]] and [[Dedekind ring]].

			== See also ==
			* [[Free object]]
			* [[Projective object]]

			==Notes==
			<references />

			==References==
			* {{cite book \| first= Iain T.\|last= Adamson \| title=Elementary Rings and Modules \| series=University Mathematical Texts \| publisher=Oliver and Boyd \| year=1972 \| isbn=0-05-002192-3 \| pages=65–66\|mr=0345993}}
			* {{cite book \|last1=Keown \|first1=R. \|title=An Introduction to Group Representation Theory \|series=Mathematics in science and engineering \|volume=116 \|year=1975 \|publisher=Academic Press \|location= \|isbn=978-0-12-404250-6 \|mr=0387387 }}
			* {{SpringerEOM \| title=Free module \| id=Free_module&oldid=13029 \| last=Govorov \| first=V. E. }}.

			==External links==
			{{PlanetMath attribution\|id=4196\|title=free vector space over a set}}

			[[Category:Module theory]]
			[[Category:Free algebraic structures]]

en>Paulthomas2: /* Ambient ionization */ Adding LAESI

2012-06-25T00:02:26Z

Ambient ionization: Adding LAESI

New page

Hi there. My title is Sophia Meagher although it is not the title on my beginning certification. What me and my family members adore is to climb but I'm thinking on beginning some thing new. Alaska is the only place I've been residing in but now I'm contemplating other options. My working day occupation is an information officer but I've currently applied for an additional one.<br><br>Here is my blog post - best psychic ([http://linkingtear.com/profile-17739/info/ http://linkingtear.com/])

Ion source - Revision history

en>Tom.Reding: Gen fixes (page/s, endash, , et al., unicodify, and/or concising wikilinks, etc.), & ref cleanup using AWB

207.255.205.112: /* Particle accelerators */

en>Paulthomas2: /* Ambient ionization */ Adding LAESI