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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 Rmodule. 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:^{[1]} 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 and or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.
If M is an Rmodule and Ann_{R}(M) = 0, then M is called a faithful module.
If S is a subset of a left R module M, then Ann(S) is a left ideal of R. The proof is straightforward: If a and b both annihilate S, then for each s in S, (a + b)s = as + bs = 0, and for any r in R, (ra)s = r(as) = r0 = 0. (A similar proof follows for subsets of right modules to show that the annihilator is a right ideal.)
If S is a submodule of M, then Ann_{R}(S) is even a twosided ideal: (ac)s = a(cs) = 0, since cs is another element of S.^{[2]}
If S is a subset of M and N is the submodule of M generated by S, then in general Ann_{R}(N) is a subset of Ann_{R}(S), but they are not necessarily equal. If R is commutative, then it is easy to check that equality holds.
M may be also viewed as a R/Ann_{R}(M)module using the action . 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_{R}(M)module, M is automatically a faithful module.
The lattice of ideals of the form where S is a subset of R comprise a complete lattice when partially ordered by inclusion. It is interesting to study rings for which this lattice (or its right counterpart) satisfy the ascending chain condition or descending chain condition.
Denote the lattice of left annihilator ideals of R as and the lattice of right annihilator ideals of R as . It is known that satisfies the A.C.C. if and only if satisfies the D.C.C., and symmetrically satisfies the A.C.C. if and only if satisfies the D.C.C. If either lattice has either of these chain conditions, then R has no infinite orthogonal sets of idempotents. (Anderson 1992, p.322) (Lam 1999)
If R is a ring for which satisfies the A.C.C. and _{R}R has finite uniform dimension, then R is called a left Goldie ring. (Lam 1999)
When R is commutative and M is an Rmodule, we may describe Ann_{R}(M) as the kernel of the action map R→End_{R}(M) determined by the adjunct map of the identity M→M along the Homtensor adjunction.
More generally, given a bilinear map of modules , the annihilator of a subset is the set of all elements in that annihilate :
Conversely, given , one can define an annihilator as a subset of .
The annihilator gives a Galois connection between subsets of and , and the associated closure operator is stronger than the span. In particular:
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 is called the orthogonal complement.
(Here we allow zero to be a zero divisor.)