In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S. Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator.
Choose a non-empty subset S of M. The annihilator of S, denoted AnnR(S), is the set of all elements r in R such that, for all s in S, rs = 0.
Subsets of right modules may be used as well, after the modification of "sr = 0" in the definition.
If the ring R can be understood from the context, the subscript R can be omitted.
or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.
If M is an R-module and AnnR(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.[2] 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.[3] 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.
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/AnnR(M)-module, M is automatically a faithful module.
is the set of prime ideals containing the subset.
[4] Given a short exact sequence of modules, the support property together with the relation with the annihilator implies More specifically, the relations If the sequence splits then the inequality on the left is always an equality.
This holds for arbitrary direct sums of modules, as Given an ideal
be a finitely generated module, then there is the relation on the support.
any finitely generated module is completely classified as the direct sum of its free part with its torsion part from the fundamental theorem of abelian groups.
Then the annihilator of a finitely generated module is non-trivial only if it is entirely torsion.
In fact the annihilator of a torsion module is isomorphic to the ideal generated by their least common multiple,
This shows the annihilators can be easily be classified over the integers.
There is a similar computation that can be done for any finitely presented module over a commutative ring
implies there exists an exact sequence, called a presentation, given by where
where S is a subset of R is a complete lattice when partially ordered by inclusion.
There is interest in studying rings for which this lattice (or its right counterpart) satisfies the ascending chain condition or descending chain condition.
Denote the lattice of left annihilator ideals of R as
satisfies the ascending chain condition if and only if
satisfies the descending chain condition, and symmetrically
If either lattice has either of these chain conditions, then R has no infinite pairwise orthogonal sets of idempotents.
and RR has finite uniform dimension, then R is called a left Goldie ring.
[8] 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.
The annihilator gives a Galois connection between subsets of
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
Given a module M over a Noetherian commutative ring R, a prime ideal of R that is an annihilator of a nonzero element of M is called an associated prime of M.