Annihilator (ring theory)

In mathematics, specifically module theory, the annihilator of a module, or a subset of a module, is a concept generalizing torsion and orthogonality. In short, for commutative rings, the annihilator of a module over a ring is the set of elements in that always act as multiplication by on . The prototypical example for an annihilator over a commutative ring can be understood by taking the quotient ring and considering it as a -module. Then, the annihilator of is the ideal since all of the act via the zero map on . This shows how the ideal can be thought of as the set of torsion elements in the base ring for the module . Also, notice that any element that isn't in will have a non-zero action on the module , implying the set can be thought of as the set of orthogonal elements to the ideal .

For noncommutative rings , there is a similar notion of the annihilator for left and right modules, called the left-annihilator and the right-annihilator.

DefinitionsEdit

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,

 

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.

PropertiesEdit

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 a 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 ringsEdit

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

Relation to supportEdit

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 sequencesEdit

Given a short exact sequence of modules

 

the support property

 [5]

together with the relation with the annihilator implies

 

Hence

 

The can be applied to computing the annihilator of a direct sum of modules, as

 

Quotient modules and annihilatorsEdit

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

 

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

 

Annihilator of quotient ringEdit

In particular, if   then the annihilator of   can be found explicitly using

 

Hence the annihilator of   is just  .

ExamplesEdit

Over the integersEdit

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 finite 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 REdit

In fact, there is a similar computation that can be done for any finite module over a commutative ring  . Recall that the definition of finiteness of   implies there exists a right-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 idealsEdit

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 A.C.C. if and only if   satisfies the D.C.C., and symmetrically   satisfies the A.C.C. if and only if   satisfies the D.C.C. If either lattice has either of these chain conditions, then R has no infinite orthogonal sets of idempotents. (Anderson & 1992, p.322) (Lam 1999)

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. (Lam 1999)

Category-theoretic description for commutative ringsEdit

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 ringsEdit

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 alsoEdit

NotesEdit

  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". stacks.math.columbia.edu. Retrieved 2020-05-13.
  5. ^ "Lemma 10.39.9 (00L3)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-13.
  6. ^ "Lemma 10.39.9 (00L3)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-13.

ReferencesEdit