Annihilator (ring theory)

In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality.

Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M. The annihilator, denoted Ann_R(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 Ann_R(x) instead of Ann_R({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 ${\displaystyle \ell .\mathrm {Ann} _{R}(S)\,}$ and ${\displaystyle r.\mathrm {Ann} _{R}(S)\,}$ or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.

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