Editing Annihilator (ring theory) (section)

==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]].