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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==
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,
:<math>\mathrm{Ann}_R(S)=\{r\in R\mid \forall s\in S, rs=0 \}\,</math>
 
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.
 
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.
 
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.
 
If ''M'' is an ''R''-module and {{nowrap|1=Ann<sub>''R''</sub>(M) = 0}}, then ''M'' is called a '''faithful module'''.
 
==Properties==
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''&nbsp;+&nbsp;''b'')''s'' = ''as''&nbsp;+&nbsp;''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.)
 
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>
 
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.
 
''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.
 
==Chain conditions on annihilator ideals==
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]].
 
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}}
 
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}}
 
==Category theoretic description for commutative rings==
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]].
 
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>:
:<math>\mbox{Ann}\,(S) := \{ n \in N \mid \forall s \in S, F(s,n) = 0\}</math>
Conversely, given <math>T \subset N</math>, one can define an annihilator as a subset of <math>M</math>.
 
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.
In particular:
* annihilators are submodules
* <math>\mbox{Span}\,(S) \leq \mbox{Ann}(\mbox{Ann}\,(S))</math>
* <math>\mbox{Ann}(\mbox{Ann}(\mbox{Ann}\,(S))) = \mbox{Ann}\,(S)</math>
 
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]].
 
==Relations to other properties of rings==
*Annihilators are used to define left [[Rickart ring]]s and [[Baer ring]]s.
*The set of (left) [[zero divisor]]s ''D''<sub>''S''</sub> of ''S'' can be written as
::<math>D_S = \bigcup_{x \in S,\, x \neq 0}{\mathrm{Ann}_R\,(x)}.</math>
(Here we allow zero to be a zero divisor.)
: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.
 
*When ''R'' is commutative, the set ''D''<sub>''R''</sub> is precisely equal to the union of the minimal prime ideals of ''R''.
 
==See also==
* [[socle (mathematics)|socle]]
 
== Notes ==
<references/>
 
==References==
*{{MathWorld|title=Annihilator|urlname=Annihilator}}
*{{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 }}
* [[Israel Nathan Herstein]] (1968) ''Noncommutative Rings'', [[Carus Mathematical Monographs]] #15, [[Mathematical Association of America]], page 3.
*{{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}}
* Richard S. Pierce. ''Associative algebras''. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN 978-0-387-90693-5
 
[[Category:Ideals]]
[[Category:Module theory]]
[[Category:Ring theory]]

Latest revision as of 09:26, 7 December 2014

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