Annihilator (ring theory)

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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.

The above definition applies also in the case of 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.


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(S), is the set of all elements r in R such that, for all s in S, rs = 0.[1] In set notation,

for all

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 "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 and 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. If R is commutative, then the equality holds.

M may be also viewed as an R/AnnR(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/AnnR(M)-module, M is automatically a faithful module.

For commutative rings[edit]

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

Relation to support[edit]

Recall that the support of a module is defined as

Then, when the module is finitely generated, there is the relation


where is the set of prime ideals containing the subset.[4]

Short exact sequences[edit]

Given a short exact sequence of modules,

the support property


together with the relation with the annihilator implies

More specifically, we have the relations

If the sequence splits then the inequality on the left is always an equality. In fact this holds for arbitrary direct sums of modules, as

Quotient modules and annihilators[edit]

Given an ideal and let be a finitely generated module, then there is the relation

on the support. Using the relation to support, this gives the relation with the annihilator[6]


Over the integers[edit]

Over 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. This is because

since the only element killing 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[edit]

In fact, there is a similar computation that can be done for any finitely presented module over a commutative ring . Recall that 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 we write each of these ideals as

then the ideal given by

presents the annihilator.

Over k[x,y][edit]

Over the commutative ring for a field , the annihilator of the module

is given by the ideal

Chain conditions on annihilator ideals[edit]

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 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. [7][8]

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.[8]

Category-theoretic description for commutative rings[edit]

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 MM 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[edit]

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.

(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.

See also[edit]


  1. ^ Pierce (1982), p. 23.
  2. ^ Proof: 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.
  3. ^ Pierce (1982), p. 23, Lemma b, item (i).
  4. ^ "Lemma 10.39.5 (00L2)—The Stacks project". Retrieved 2020-05-13.
  5. ^ "Lemma 10.39.9 (00L3)—The Stacks project". Retrieved 2020-05-13.
  6. ^ "Lemma 10.39.9 (00L3)—The Stacks project". Retrieved 2020-05-13.
  7. ^ Anderson & Fuller 1992, p. 322.
  8. ^ a b Lam 1999.