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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==
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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]]

Latest revision as of 19:15, 5 January 2015


This particular next step to particular games success is that may it produces the picture that its a multi player game. I believe its a fantasy for the you dont do all that's necessary directly with yet player. You dont fight and explore in unison like you would in Wow, of play onto another player even during with a turn-by-turn purpose comparable to Chess. Any time you raid another players village, which in turn player is offline and you could at how the same time just develop into raiding a random computer-generated village.

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In cases where you're playing a quest online, and you cost across another player of which seems to be discouraging other players (or you, in particular) intentionally, never will take it personally. This is called "Griefing," and it's the video game equivalent of Internet trolling. Griefers are pretty much out for negative attention, and you give them what they're looking designed for if you interact these. Don't get emotionally used in what's happening on top of that simply try to overlook it.

Supercell has absolutely considerable and explained the steps akin to Association Wars, the once again appear passion in Deviate of Clans. As your name recommends, a correlation war is often one particular strategic battle amid multiple clans. It just takes abode over the introduction of two canicule -- the actual alertness day plus your action day -- while the acceptable association along with a ample boodle bonus; although, every association affiliate to who makes acknowledged attacks after a association war additionally produces some benefit loot.

A person's war abject is agnate in your approved village, except that your war abject will not gain resources. Barrio when your warfare abject cannot be anon improved possibly rearranged, as it on it's own mimics this adjustment and then accomplished completed advancement amounts of your apple throughout the time of alertness day. Struggle bases additionally never charges to take their gear rearmed, defenses reloaded maybe characters healed, as these kinds of products are consistently ready. The association alcazar with your war abject penalty be abounding alone to the one in the whole village.

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