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 rightannihilator, of a right module is a right ideal.
Definitions
[edit]Let R be a ring, and let M be a left Rmodule. Choose a nonempty 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, 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 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 Rmodule, 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 Rmodule and Ann_{R}(M) = 0, then M is called a faithful module.
Properties
[edit]If S is a subset of a left Rmodule M, then Ann(S) is a left ideal of R.^{[2]}
If S is a submodule of M, then Ann_{R}(S) is even a twosided 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 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 an R/Ann_{R}(M)module using the action . Incidentally, it is not always possible to make an Rmodule into an R/Imodule this way, but if the ideal I is a subset of the annihilator of M, then this action is welldefined. Considered as an R/Ann_{R}(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
 ^{[5]}
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]}
Examples
[edit]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 nontrivial 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 is a complete lattice when partially ordered by inclusion. It is interesting to study 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 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 _{R}R has finite uniform dimension, then R is called a left Goldie ring.^{[8]}
Categorytheoretic description for commutative rings
[edit]When R is commutative and M is an Rmodule, we may describe Ann_{R}(M) as the kernel of the action map R → End_{R}(M) determined by the adjunct map of the identity M → M along the Homtensor 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.
 Annihilators are used to define left Rickart rings and Baer rings.
 The set of (left) zero divisors D_{S} of S can be written as
 (Here we allow zero to be a zero divisor.)
 In particular D_{R} is the set of (left) zero divisors of R taking S = R and R acting on itself as a left Rmodule.
 When R is commutative and Noetherian, the set is precisely equal to the union of the associated primes of the Rmodule R.
See also
[edit]Notes
[edit] ^ Pierce (1982), p. 23.
 ^ 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.
 ^ Pierce (1982), p. 23, Lemma b, item (i).
 ^ "Lemma 10.39.5 (00L2)—The Stacks project". stacks.math.columbia.edu. Retrieved 20200513.
 ^ "Lemma 10.39.9 (00L3)—The Stacks project". stacks.math.columbia.edu. Retrieved 20200513.
 ^ "Lemma 10.39.9 (00L3)—The Stacks project". stacks.math.columbia.edu. Retrieved 20200513.
 ^ Anderson & Fuller 1992, p. 322.
 ^ ^{a} ^{b} Lam 1999.
References
[edit] Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: SpringerVerlag, pp. x+376, doi:10.1007/9781461244189, ISBN 0387978453, MR 1245487
 Israel Nathan Herstein (1968) Noncommutative Rings, Carus Mathematical Monographs #15, Mathematical Association of America, page 3.
 Lam, Tsit Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York: SpringerVerlag, pp. 228–232, doi:10.1007/9781461205258, ISBN 9780387984285, MR 1653294
 Richard S. Pierce. Associative algebras. Graduate Texts in Mathematics, Vol. 88, SpringerVerlag, 1982, ISBN 9780387906935