Editing Annihilator (ring theory) (section)

==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''.