Annihilator (ring theory)


In mathematics, the annihilator of a subset 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. For example, if is a commutative ring and is an ideal of, we can consider the quotient ring to be an -module. Then, the annihilator of is the ideal, since all of the act via the zero map on.
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.
The above definition applies also in the case of noncommutative rings, a subset of a left module has a left annihilator, which is a left ideal, and a subset of a right module has a right annihilator, which is a right ideal. If is a module, then the annihilator is always a two-sided ideal, regardless of whether the module is a left module or a right module.

Definitions

Let R be a ring, and let M be a left R-module. Choose a non-empty subset S of M. The annihilator of S, denoted AnnR, is the set of all elements r in R such that, for all s in S,. In set notation,
It is the set of all elements of R that "annihilate" S. Subsets of right modules may be used as well, after the modification of "" in the definition.
The annihilator of a single element x is usually written AnnR instead of AnnR. 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 R-module and, then M is called a faithful module.

Properties

If S is a subset of a left R-module M, then Ann is a left ideal of R. If S is not just a subset but also a submodule of M, then AnnR is moreover a two-sided ideal: s = a = 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 is a subset of AnnR, but they are not necessarily equal. If R is commutative, then the equality holds.
An R-module M may be also viewed as an R/AnnR-module using the action. 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. The module M is always faithful when considered as an R/AnnR-module.

For commutative rings

Throughout this section, let be a commutative ring and an -module.

Relation to support

The support of a module is defined as
When the module is finitely generated, the support of is exactly the set of prime ideals containing.

Short exact sequences

Given a short exact sequence of modules,
we have the relations
If the sequence splits, so that, then the inequality on the left is always an equality. Indeed, this holds for arbitrary direct sums of modules:
In the special case that and for some ideal, we have the relation.

Examples

Over the integers

Over, any finitely generated module is completely classified, by the fundamental theorem of abelian groups, as a direct sum between its free part and its torsion part. Therefore, if the annihilator of a finitely generated module is non-trivial, it must be exactly equal to the torsion part of the module. This is because
since the only element annihilating each of the is. For example, the annihilator of is
the ideal generated by. 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.

Over a commutative ring ''R''

There is a similar computation that can be done for any finitely presented module over a commutative ring. The definition of finite presentedness of implies there exists an exact sequence, called a presentation, given by
where is in. Writing explicitly as a matrix gives it as
hence has the direct sum decomposition
If each of these ideals is written as
then the ideal given by
presents the annihilator.

Over ''k''''x'',''y''

Over the commutative ring for a field, the annihilator of the module
is given by the ideal

Chain conditions on annihilator ideals

The lattice of ideals of the form 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 satisfies 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 ascending chain condition if and only if satisfies the descending chain condition, and symmetrically satisfies the ascending chain condition if and only if satisfies the descending chain condition. If either lattice has either of these chain conditions, then R has no infinite pairwise orthogonal sets of idempotents.
If R is a ring for which satisfies the A.C.C. and RR has finite uniform dimension, then R is called a left Goldie ring.

Category-theoretic description for commutative rings

When is commutative and is an -module, we may describe as the kernel of the action map determined by the adjunct map of the identity along the Hom-tensor 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:
  • annihilators are submodules
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.

Relations to other properties of rings

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.