In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality.
Contents
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 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
If M is an R-module and AnnR(M) = 0, then M is called a faithful module.
Properties
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 AnnR(S) is even a two-sided ideal: (ac)s = a(cs) = 0, since cs is another element of S.
If S is a subset of M and N is the submodule of M generated by S, then in general AnnR(N) is a subset of AnnR(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/AnnR(M)-module using the action
Chain conditions on annihilator ideals
The lattice of ideals of the form
Denote the lattice of left annihilator ideals of R as
If R is a ring for which
Category-theoretic description for commutative rings
When R is commutative and M is an R-module, we may describe AnnR(M) as the kernel of the action map R → EndR(M) determined by the adjunct map of the identity M → M along the Hom-tensor adjunction.
More generally, given a bilinear map of modules
Conversely, given
The annihilator gives a Galois connection between subsets of
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
Relations to other properties of rings
(Here we allow zero to be a zero divisor.)
In particular DR is the set of (left) zero divisors of R taking S = R and R acting on itself as a left R-module.