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 an element of . 
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 noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal.
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 Ann"R"("S"), 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" (the elements for which "S" is a torsion set). 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 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 formula_2 and formula_3 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("S") is a left ideal of "R".
If "S" is a submodule of "M", then Ann"R"("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 Ann"R"("N") is a subset of Ann"R"("S"), but they are not necessarily equal. If "R" is commutative, then the equality holds.
"M" may be also viewed as a "R"/Ann"R"("M")-module using the action formula_4. 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.
For commutative rings.
Throughout this section, let formula_5 be a commutative ring and formula_6 a finitely generated (for short, finite) formula_5-module.
Relation to support.
Recall that the support of a module is defined as
Then, when the module is finitely generated, there is the relation
where formula_10 is the set of prime ideals containing the subset.
Short exact sequences.
Given a short exact sequence of modules,
the support property
together with the relation with the annihilator implies
Hence,
This can be applied to computing the annihilator of a direct sum of modules, as
Quotient modules and annihilators.
Given an ideal formula_16 and let formula_6 be a finite module, then there is the relationformula_18on the support. Using the relation to support, this gives the relation with the annihilatorformula_19
Examples.
Over the integers.
Over formula_20 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 finite module is non-trivial only if it is entirely torsion. This is becauseformula_21since the only element killing each of the formula_20 is formula_23. For example, the annihilator of formula_24 isformula_25the ideal generated by formula_26. In fact the annihilator of a torsion moduleformula_27is isomorphic to the ideal generated by their least common multiple, formula_28. This shows the annihilators can be easily be classified over the integers.
Over a commutative ring "R".
In fact, there is a similar computation that can be done for any finite module over a commutative ring formula_5. Recall that the definition of finiteness of formula_6 implies there exists a right-exact sequence, called a presentation, given byformula_31where formula_32 is in formula_33. Writing formula_32 explicitly as a matrix gives it asformula_35hence formula_6 has the direct sum decompositionformula_37If we write each of these ideals asformula_38then the ideal formula_39 given byformula_40presents the annihilator.
Over "k"["x","y"].
Over the commutative ring formula_41 for a field formula_42, the annihilator of the moduleformula_43is given by the idealformula_44
Chain conditions on annihilator ideals.
The lattice of ideals of the form formula_2 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 formula_46 and the lattice of right annihilator ideals of "R" as formula_47. It is known that formula_46 satisfies the A.C.C. if and only if formula_47 satisfies the D.C.C., and symmetrically formula_47 satisfies the A.C.C. if and only if formula_46 satisfies the D.C.C. If either lattice has either of these chain conditions, then "R" has no infinite orthogonal sets of idempotents. 
If "R" is a ring for which formula_46 satisfies the A.C.C. and "R""R" has finite uniform dimension, then "R" is called a left Goldie ring. 
Category-theoretic description for commutative rings.
When "R" is commutative and "M" is an "R"-module, we may describe Ann"R"("M") 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 formula_53, the annihilator of a subset formula_54 is the set of all elements in formula_55 that annihilate formula_56:
Conversely, given formula_58, one can define an annihilator as a subset of formula_6.
The annihilator gives a Galois connection between subsets of formula_6 and formula_55, 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 formula_64 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".