Definitions
Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M. The annihilator, denoted AnnR(S), of S is the set of all elements r in R such that for each s in S, rs = 0: In set notation,
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 "sr =0" in the definition.
The annihilator of a single element x is usually written AnnR(x) instead of AnnR({x}). If the ring R can be understood from the context, the subscript R can be omitted.
Since S may be taken to be a subset of R itself, and R is both a right and a left R module, the notation must be modified slightly to indicate the left or right side. Usually and or some similar subscript scheme are used to indicate the left and right annihilators, if necessary.
If M is an R module and AnnR(M) =0, then M is called a faithful module.
Read more about this topic: Annihilator (ring Theory)
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