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| | Luke Bryan is really a superstar while in the producing and also the occupation expansion initial next to his third hotel recording, And , will be the proof. He burst to the picture in 2000 regarding his amusing blend of straight down-residence ease of access, film star good seems and lyrics, is placed t in the main way. The new recor in the nation graph or chart and #2 in the burst graphs, earning it the 2nd highest very first in those days of 2003 for a region performer. <br><br>The kid of your , is aware perseverance and perseverance are key elements when it comes to a prosperous profession- . His initially record, Keep Me, created the luke bryan book ([http://lukebryantickets.omarfoundation.org omarfoundation.org]) best reaches “All My [http://lukebryantickets.asiapak.net tickets to see luke bryan] Buddies “Country and Say” Guy,” whilst his work, Doin’ Point, located the vocalist-three directly No. 3 single men and women: More concert dates for luke bryan ([http://www.ffpjp24.org simply click the next document]) Phoning Can be a Very good Point.”<br><br>In the slip of 2002, Concert tour: Bryan & which had an amazing list of [http://www.ladyhawkshockey.org discount concert tickets] , which include Urban. “It’s almost like you are receiving a authorization to look one stage further, states those designers that have been an element of the Concert tourover in a greater degree of artists.” It twisted as the best trips in the twenty-calendar year historical past.<br><br>Also visit my web site: [http://www.cinemaudiosociety.org luke bryan concert dates for 2014] |
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| In [[mathematics]], specifically [[module theory]], the '''annihilator''' of a set is a concept generalizing [[Torsion (algebra)|torsion]] and [[Orthogonality#Euclidean_vector_spaces|orthogonality]].
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| ==Definitions==
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| Let ''R'' be a [[ring (mathematics)|ring]], and let ''M'' be a left ''R''-[[module (mathematics)|module]]. Choose a nonempty subset ''S'' of ''M''. The '''annihilator''', denoted Ann<sub>''R''</sub>(''S''), of ''S'' is the set of all elements ''r'' in ''R'' such that for each ''s'' in ''S'', {{nowrap|1=''rs'' = 0}}:<ref>Pierce (1982), p. 23.</ref> In set notation,
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| :<math>\mathrm{Ann}_R(S)=\{r\in R\mid \forall s\in S, rs=0 \}\,</math>
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| It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is torsion). Subsets of right modules may be used as well, after the modification of "{{nowrap|1=''sr'' = 0}}" in the definition.
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| The annihilator of a single element ''x'' is usually written Ann<sub>''R''</sub>(''x'') instead of Ann<sub>''R''</sub>({''x''}). If the ring ''R'' can be understood from the context, the subscript ''R'' can be omitted. | |
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| Since ''R'' is a module over itself, ''S'' may be taken to be a subset of ''R'' itself, and since ''R'' is both a right and a left ''R'' module, the notation must be modified slightly to indicate the left or right side. Usually <math>\ell.\mathrm{Ann}_R(S)\,</math> and <math>r.\mathrm{Ann}_R(S)\,</math> or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.
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| If ''M'' is an ''R''-module and {{nowrap|1=Ann<sub>''R''</sub>(M) = 0}}, then ''M'' is called a '''faithful module'''.
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| ==Properties==
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| If ''S'' is a subset of a left ''R'' module ''M'', then Ann(''S'') is a [[ideal (ring theory)#Definitions|left ideal]] of ''R''. The proof is straightforward: If ''a'' and ''b'' both annihilate ''S'', then for each ''s'' in ''S'', (''a'' + ''b'')''s'' = ''as'' + ''bs'' = 0, and for any ''r'' in ''R'', (''ra'')''s'' = ''r''(''as'') = ''r''0 = 0. (A similar proof follows for subsets of right modules to show that the annihilator is a right ideal.)
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| If ''S'' is a submodule of ''M'', then Ann<sub>''R''</sub>(''S'') is even a two-sided ideal: (''ac'')''s'' = ''a''(''cs'') = 0, since ''cs'' is another element of ''S''.<ref>Pierce (1982), p. 23, Lemma b, item (i).</ref>
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| If ''S'' is a subset of ''M'' and ''N'' is the submodule of ''M'' generated by ''S'', then in general Ann<sub>''R''</sub>(''N'') is a subset of Ann<sub>''R''</sub>(''S''), but they are not necessarily equal. If ''R'' is commutative, then it is easy to check that equality holds.
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| ''M'' may be also viewed as a ''R''/Ann<sub>''R''</sub>(''M'')-module using the action <math>\overline{r}m:=rm\,</math>. Incidentally, it is not always possible to make an ''R'' module into an ''R''/''I'' module this way, but if the ideal ''I'' is a subset of the annihilator of ''M'', then this action is well defined. Considered as an ''R''/Ann<sub>''R''</sub>(''M'')-module, ''M'' is automatically a faithful module.
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| ==Chain conditions on annihilator ideals==
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| The lattice of ideals of the form <math>\ell.\mathrm{Ann}_R(S)\,</math> where ''S'' is a subset of ''R'' comprise a [[complete lattice]] when partially ordered by inclusion. It is interesting to study rings for which this lattice (or its right counterpart) satisfy the [[ascending chain condition]] or [[descending chain condition]].
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| Denote the lattice of left annihilator ideals of ''R'' as <math>\mathcal{LA}\,</math> and the lattice of right annihilator ideals of ''R'' as <math>\mathcal{RA}\,</math>. It is known that <math>\mathcal{LA}\,</math> satisfies the A.C.C. if and only if <math>\mathcal{RA}\,</math> satisfies the D.C.C., and symmetrically <math>\mathcal{RA}\,</math> satisfies the A.C.C. if and only if <math>\mathcal{LA}\,</math> satisfies the D.C.C. If either lattice has either of these chain conditions, then ''R'' has no infinite orthogonal sets of [[idempotent element|idempotent]]s. {{harv|Anderson|1992, p.322}} {{harv|Lam|1999}}
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| If ''R'' is a ring for which <math>\mathcal{LA}\,</math> satisfies the A.C.C. and <sub>''R''</sub>''R'' has finite [[Uniform module#Uniform dimension of a module|uniform dimension]], then ''R'' is called a left [[Goldie ring]]. {{harv|Lam|1999}}
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| ==Category theoretic description for commutative rings==
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| When ''R'' is commutative and ''M'' is an ''R''-module, we may describe ''Ann''<sub>''R''</sub>(''M'') as the kernel of the action map ''R''→''End''<sub>''R''</sub>(''M'') determined by the [[adjunction (category theory)|adjunct map]] of the identity ''M''→''M'' along the [[Hom-tensor adjunction]].
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| More generally, given a [[bilinear map]] of modules <math>F\colon M \times N \to P</math>, the annihilator of a subset <math>S \subset M</math> is the set of all elements in <math>N</math> that annihilate <math>S</math>:
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| :<math>\mbox{Ann}\,(S) := \{ n \in N \mid \forall s \in S, F(s,n) = 0\}</math>
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| Conversely, given <math>T \subset N</math>, one can define an annihilator as a subset of <math>M</math>.
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| The annihilator gives a [[Galois connection]] between subsets of <math>M</math> and <math>N</math>, and the associated [[closure operator]] is stronger than the span.
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| In particular:
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| * annihilators are submodules
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| * <math>\mbox{Span}\,(S) \leq \mbox{Ann}(\mbox{Ann}\,(S))</math>
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| * <math>\mbox{Ann}(\mbox{Ann}(\mbox{Ann}\,(S))) = \mbox{Ann}\,(S)</math>
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| An important special case is in the presence of a [[nondegenerate form]] on a vector space, particularly an [[inner product]]: then the annihilator associated to the map <math>V \times V \to K</math> is called the [[orthogonal complement]].
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| ==Relations to other properties of rings==
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| *Annihilators are used to define left [[Rickart ring]]s and [[Baer ring]]s.
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| *The set of (left) [[zero divisor]]s ''D''<sub>''S''</sub> of ''S'' can be written as
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| ::<math>D_S = \bigcup_{x \in S,\, x \neq 0}{\mathrm{Ann}_R\,(x)}.</math>
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| (Here we allow zero to be a zero divisor.)
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| :In particular ''D<sub>R</sub>'' is the set of (left) zero divisors of ''R'' taking ''S'' = ''R'' and ''R'' acting on itself as a left ''R''-module.
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| *When ''R'' is commutative, the set ''D''<sub>''R''</sub> is precisely equal to the union of the minimal prime ideals of ''R''.
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| ==See also==
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| * [[socle (mathematics)|socle]]
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| == Notes ==
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| <references/>
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| ==References==
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| *{{MathWorld|title=Annihilator|urlname=Annihilator}}
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| *{{citation |author1=Anderson, Frank W. |author2=Fuller, Kent R. |title=Rings and categories of modules |series=Graduate Texts in Mathematics |volume=13 |edition=2 |publisher=Springer-Verlag |place=New York |year=1992 |pages=x+376 |isbn=0-387-97845-3 |mr=1245487 }}
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| * [[Israel Nathan Herstein]] (1968) ''Noncommutative Rings'', [[Carus Mathematical Monographs]] #15, [[Mathematical Association of America]], page 3.
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999|pages=228–232}}
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| * Richard S. Pierce. ''Associative algebras''. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN 978-0-387-90693-5
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| [[Category:Ideals]]
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| [[Category:Module theory]]
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| [[Category:Ring theory]]
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In the slip of 2002, Concert tour: Bryan & which had an amazing list of discount concert tickets , which include Urban. “It’s almost like you are receiving a authorization to look one stage further, states those designers that have been an element of the Concert tourover in a greater degree of artists.” It twisted as the best trips in the twenty-calendar year historical past.
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