Editing Annihilator (ring theory)

{{Short description|Ideal that maps to zero a subset of a module}}
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{{Refimprove|date=January 2010}}
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In [[mathematics]], the '''annihilator''' of a [[subset]] {{mvar|S}} of a [[module (mathematics)|module]] over a [[ring (mathematics)|ring]] is the [[ideal (ring theory)|ideal]] formed by the elements of the ring that give always zero when multiplied by each element of {{mvar|S}}. 

Over an [[integral domain]], a module that has a nonzero annihilator is a [[torsion module]], and a [[finitely generated module|finitely generated]] torsion module has a nonzero annihilator.

The above definition applies also in the case of [[Noncommutative ring|noncommutative rings]], where the '''left annihilator''' of a left module is a left ideal, and the  '''right-annihilator''', of a right module is a right ideal.

==Definitions==
Let ''R'' be a [[ring (mathematics)|ring]], and let ''M'' be a left ''R''-[[module (mathematics)|module]].  Choose a [[empty set|non-empty]] subset ''S'' of ''M''.  The ''annihilator'' of ''S'', denoted Ann<sub>''R''</sub>(''S''), is the set of all elements ''r'' in ''R'' such that, for all ''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 rs = 0</math> for all <math> s\in S \}</math>

It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is a torsion set).  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 left [[Ideal (ring theory)#Definitions|ideal]] of ''R''.<ref>Proof: 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.</ref>

If ''S'' is a [[Module_(mathematics)#Submodules_and_homomorphisms|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 ring|commutative]], then the equality holds.

''M'' may be also viewed as an ''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.

=== For commutative rings ===
Throughout this section, let <math>R</math> be a commutative ring and <math>M</math> a [[finitely generated module|finitely generated]] <math>R</math>-module.

==== Relation to support ====
The [[support of a module]] is defined as
:<math>\operatorname{Supp}M = \{ \mathfrak{p} \in \operatorname{Spec}R \mid M_\mathfrak{p} \neq 0 \}.</math>
Then, when the module is finitely generated, there is the relation
:<math>V(\operatorname{Ann}_R(M)) = \operatorname{Supp}M</math>,
where <math>V(\cdot)</math> is the set of [[prime ideal]]s containing the subset.<ref>{{Cite web|title=Lemma 10.39.5 (00L2)—The Stacks project|url=https://stacks.math.columbia.edu/tag/00L2|website=stacks.math.columbia.edu|access-date=2020-05-13}}</ref>

==== Short exact sequences ====
Given a [[short exact sequence]] of modules,
:<math>0 \to M' \to M \to M'' \to 0,</math>
the support property
:<math>\operatorname{Supp}M = \operatorname{Supp}M' \cup \operatorname{Supp}M'',</math><ref>{{Cite web|title=Lemma 10.39.9 (00L3)—The Stacks project|url=https://stacks.math.columbia.edu/tag/00L3|website=stacks.math.columbia.edu|access-date=2020-05-13}}</ref>
together with the relation with the annihilator implies
:<math>V(\operatorname{Ann}_R(M)) = V(\operatorname{Ann}_R(M')) \cup V(\operatorname{Ann}_R(M'')).</math>
More specifically, the relations
:<math>\operatorname{Ann}_R(M') \cap \operatorname{Ann}_R(M'') \supseteq \operatorname{Ann}_R(M) \supseteq \operatorname{Ann}_R(M') \operatorname{Ann}_R(M''). </math>
If the sequence splits then the inequality on the left is always an equality. This holds for arbitrary [[direct sum of modules|direct sums]] of modules, as
:<math>\operatorname{Ann}_R\left( \bigoplus_{i\in I} M_i \right) = \bigcap_{i\in I} \operatorname{Ann}_R(M_i).</math>

==== Quotient modules and annihilators ====
Given an ideal <math>I \subseteq R</math> and let <math>M</math> be a finitely generated module, then there is the relation
:<math>\text{Supp}(M/IM) = \operatorname{Supp}M \cap V(I)</math>
on the support. Using the relation to support, this gives the relation with the annihilator<ref>{{Cite web|title=Lemma 10.39.9 (00L3)—The Stacks project|url=https://stacks.math.columbia.edu/tag/00L3|website=stacks.math.columbia.edu|access-date=2020-05-13}}</ref>
:<math>V(\text{Ann}_R(M/IM)) = V(\text{Ann}_R(M)) \cap V(I).</math>

== Examples ==

=== Over the integers ===
Over <math>\mathbb{Z}</math> any finitely generated module is completely classified as the direct sum of its [[free module|free]] part with its torsion part from the fundamental theorem of abelian groups. Then the annihilator of a finitely generated module is non-trivial only if it is entirely torsion. This is because
:<math>\text{Ann}_{\mathbb{Z}}(\mathbb{Z}^{\oplus k}) = \{ 0 \} = (0)</math>
since the only element killing each of the <math>\mathbb{Z}</math> is <math>0</math>. For example, the annihilator of <math>\mathbb{Z}/2 \oplus \mathbb{Z}/3</math> is
:<math>\text{Ann}_\mathbb{Z}(\mathbb{Z}/2 \oplus \mathbb{Z}/3) = (6) = (\text{lcm}(2,3)),</math>
the ideal generated by <math>(6)</math>. In fact the annihilator of a torsion module
:<math>M \cong \bigoplus_{i=1}^n (\mathbb{Z}/a_i)^{\oplus k_i}</math>
is [[isomorphic]] to the ideal generated by their [[least common multiple]], <math>(\operatorname{lcm}(a_1, \ldots, a_n))</math>. This shows the annihilators can be easily be classified over the integers.

=== Over a commutative ring ''R'' ===
There is a similar computation that can be done for any [[finitely presented module]] over a commutative ring <math>R</math>. The definition of finite presentedness of <math>M</math> implies there exists an exact sequence, called a presentation, given by
:<math>R^{\oplus l} \xrightarrow{\phi} R^{\oplus k} \to M \to 0</math>
where <math>\phi</math> is in <math>\text{Mat}_{k,l}(R)</math>. Writing <math>\phi</math> explicitly as a [[matrix (mathematics)|matrix]] gives it as
:<math>\phi = \begin{bmatrix}
\phi_{1,1} & \cdots & \phi_{1,l} \\
\vdots &  & \vdots \\
\phi_{k,1} & \cdots & \phi_{k,l}
\end{bmatrix};</math>
hence <math>M</math> has the direct sum decomposition
:<math>M = \bigoplus_{i=1}^k \frac{R}{(\phi_{i,1}(1), \ldots, \phi_{i,l}(1))}</math>
If each of these ideals is written as
:<math>I_i = (\phi_{i,1}(1), \ldots, \phi_{i,l}(1))</math>
then the ideal <math>I</math> given by
:<math>V(I) = \bigcup^{k}_{i=1}V(I_i)</math>
presents the annihilator.

=== Over ''k''[''x'',''y''] ===
Over the commutative ring <math>k[x,y]</math> for a [[field (mathematics)|field]] <math>k</math>, the annihilator of the module
:<math>M = \frac{k[x,y]}{(x^2 - y)} \oplus \frac{k[x,y]}{(y - 3)}</math>
is given by the ideal
:<math>\text{Ann}_{k[x,y]}(M) = ((x^2 - y)(y - 3)).</math>

==Chain conditions on annihilator ideals==
The [[Lattice (order)|lattice]] of ideals of the form <math>\ell.\!\mathrm{Ann}_R(S)</math> where ''S'' is a subset of ''R'' is a [[complete lattice]] when [[partially ordered]] by [[subset|inclusion]].  There is interest in studying rings for which this lattice (or its right counterpart) satisfies 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 ascending chain condition [[if and only if]] <math>\mathcal{RA}\,</math> satisfies the descending chain condition, and symmetrically <math>\mathcal{RA}\,</math> satisfies the ascending chain condition if and only if <math>\mathcal{LA}\,</math> satisfies the descending chain condition. If either lattice has either of these chain conditions, then ''R'' has no infinite pairwise orthogonal sets of [[idempotent (ring theory)|idempotents]]. {{sfn|Anderson|Fuller|1992|p=322}}{{sfn|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]].{{sfn|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 (algebra)|kernel]] of the action map {{nowrap|''R'' → End<sub>''R''</sub>(''M'')}} determined by the [[Adjunction (category theory)|adjunct map]] of the [[identity map|identity]] {{nowrap|''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 \subseteq M</math> is the set of all elements in <math>N</math> that annihilate <math>S</math>:
:<math>\operatorname{Ann}(S) := \{ n \in N \mid \forall s \in S: F(s,n) = 0 \} .</math>
Conversely, given <math>T \subseteq 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>\operatorname{Span}S \leq \operatorname{Ann}(\operatorname{Ann}(S))</math>
* <math>\operatorname{Ann}(\operatorname{Ann}(\operatorname{Ann}(S))) = \operatorname{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==
Given a module ''M'' over a [[Noetherian ring|Noetherian]] commutative ring ''R'', a prime ideal of ''R'' that is an annihilator of a nonzero element of ''M'' is called an [[associated prime]] of ''M''.

*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 \setminus \{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 and Noetherian, the set <math>D_R</math> is precisely equal to the [[Union (set theory)|union]] of the [[associated prime]]s of the ''R''-module ''R''.

==See also==
*[[Faltings' annihilator theorem]]
* [[socle (mathematics)|Socle]]
*[[Support of a module]]

==Notes==
 <references/>

==References==
*{{citation  |last1=Anderson |first1=Frank W.   |last2=Fuller |first2=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 |doi=10.1007/978-1-4612-4418-9}}
* [[Israel Nathan Herstein]] (1968) ''Noncommutative Rings'', [[Carus Mathematical Monographs]] #15, [[Mathematical Association of America]], page 3.
*{{Citation | last1=Lam | first1=Tsit Yuen |author-link = Tsit Yuen Lam| 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| volume=189 |pages=228–232 | doi=10.1007/978-1-4612-0525-8}}
* Richard S. Pierce. ''Associative algebras''. Graduate Texts in Mathematics, Vol. 88, Springer-Verlag, 1982, {{ISBN|978-0-387-90693-5}}

[[Category:Ideals (ring theory)]]
[[Category:Module theory]]
[[Category:Ring theory]]