Annihilator (ring theory)
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 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 AnnR(S), is the set of all elements r in R such that, for all s in S, rs = 0.[1] In set notation,
- [math]\displaystyle{ \mathrm{Ann}_R(S)=\{r\in R\mid rs = 0 }[/math] for all [math]\displaystyle{ s\in S \} }[/math]
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 [math]\displaystyle{ \ell.\!\mathrm{Ann}_R(S)\, }[/math] and [math]\displaystyle{ r.\!\mathrm{Ann}_R(S)\, }[/math] 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.
Properties
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 [math]\displaystyle{ \overline{r}m:=rm\, }[/math]. 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
Throughout this section, let [math]\displaystyle{ R }[/math] be a commutative ring and [math]\displaystyle{ M }[/math] a finitely generated (for short, finite) [math]\displaystyle{ R }[/math]-module.
Relation to support
Recall that the support of a module is defined as
- [math]\displaystyle{ \operatorname{Supp}M = \{ \mathfrak{p} \in \operatorname{Spec}R \mid M_\mathfrak{p} \neq 0 \}. }[/math]
Then, when the module is finitely generated, there is the relation
- [math]\displaystyle{ V(\operatorname{Ann}_R(M)) = \operatorname{Supp}M }[/math],
where [math]\displaystyle{ V(\cdot) }[/math] is the set of prime ideals containing the subset.[4]
Short exact sequences
Given a short exact sequence of modules,
- [math]\displaystyle{ 0 \to M' \to M \to M'' \to 0, }[/math]
the support property
- [math]\displaystyle{ \operatorname{Supp}M = \operatorname{Supp}M' \cup \operatorname{Supp}M'', }[/math][5]
together with the relation with the annihilator implies
- [math]\displaystyle{ V(\operatorname{Ann}_R(M)) = V(\operatorname{Ann}_R(M')) \cup V(\operatorname{Ann}_R(M'')). }[/math]
More specifically, we have the relations
- [math]\displaystyle{ \operatorname{Ann}_R(M') \cap \operatorname{Ann}_R(M'') \supseteq \operatorname{Ann}_R(M) \supseteq \operatorname{Ann}_R(M') \operatorname{Ann}_R(M''). }[/math]
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
- [math]\displaystyle{ \operatorname{Ann}_R\left( \bigoplus_{i\in I} M_i \right) = \bigcap_{i\in I} \operatorname{Ann}_R(M_i). }[/math]
Quotient modules and annihilators
Given an ideal [math]\displaystyle{ I \subseteq R }[/math] and let [math]\displaystyle{ M }[/math] be a finite module, then there is the relation
- [math]\displaystyle{ \text{Supp}(M/IM) = \operatorname{Supp}M \cap V(I) }[/math]
on the support. Using the relation to support, this gives the relation with the annihilator[6]
- [math]\displaystyle{ V(\text{Ann}_R(M/IM)) = V(\text{Ann}_R(M)) \cap V(I). }[/math]
Examples
Over the integers
Over [math]\displaystyle{ \mathbb{Z} }[/math] 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
- [math]\displaystyle{ \text{Ann}_{\mathbb{Z}}(\mathbb{Z}^{\oplus k}) = \{ 0 \} = (0) }[/math]
since the only element killing each of the [math]\displaystyle{ \mathbb{Z} }[/math] is [math]\displaystyle{ 0 }[/math]. For example, the annihilator of [math]\displaystyle{ \mathbb{Z}/2 \oplus \mathbb{Z}/3 }[/math] is
- [math]\displaystyle{ \text{Ann}_\mathbb{Z}(\mathbb{Z}/2 \oplus \mathbb{Z}/3) = (6) = (\text{lcm}(2,3)), }[/math]
the ideal generated by [math]\displaystyle{ (6) }[/math]. In fact the annihilator of a torsion module
- [math]\displaystyle{ M \cong \bigoplus_{i=1}^n (\mathbb{Z}/a_i)^{\oplus k_i} }[/math]
is isomorphic to the ideal generated by their least common multiple, [math]\displaystyle{ (\operatorname{lcm}(a_1, \ldots, a_n)) }[/math]. 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 [math]\displaystyle{ R }[/math]. Recall that the definition of finiteness of [math]\displaystyle{ M }[/math] implies there exists a right-exact sequence, called a presentation, given by
- [math]\displaystyle{ R^{\oplus l} \xrightarrow{\phi} R^{\oplus k} \to M \to 0 }[/math]
where [math]\displaystyle{ \phi }[/math] is in [math]\displaystyle{ \text{Mat}_{k,l}(R) }[/math]. Writing [math]\displaystyle{ \phi }[/math] explicitly as a matrix gives it as
- [math]\displaystyle{ \phi = \begin{bmatrix} \phi_{1,1} & \cdots & \phi_{1,n} \\ \vdots & & \vdots \\ \phi_{n,1} & \cdots & \phi_{n,n} \end{bmatrix} }[/math]
hence [math]\displaystyle{ M }[/math] has the direct sum decomposition
- [math]\displaystyle{ M = \bigoplus_{i=1}^k \frac{R}{(\phi_{i,1}(1), \ldots, \phi_{i,n}(1))} }[/math]
If we write each of these ideals as
- [math]\displaystyle{ I_i = (\phi_{i,1}(1), \ldots, \phi_{i,n}(1)) }[/math]
then the ideal [math]\displaystyle{ I }[/math] given by
- [math]\displaystyle{ V(I) = \bigcup^{n}_{i=1}V(I_i) }[/math]
presents the annihilator.
Over k[x,y]
Over the commutative ring [math]\displaystyle{ k[x,y] }[/math] for a field [math]\displaystyle{ k }[/math], the annihilator of the module
- [math]\displaystyle{ M = \frac{k[x,y]}{(x^2 - y)} \oplus \frac{k[x,y]}{(y - 3)} }[/math]
is given by the ideal
- [math]\displaystyle{ \text{Ann}_{k[x,y]}(M) = ((x^2 - y)(y - 3)). }[/math]
Chain conditions on annihilator ideals
The lattice of ideals of the form [math]\displaystyle{ \ell.\!\mathrm{Ann}_R(S) }[/math] 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 [math]\displaystyle{ \mathcal{LA}\, }[/math] and the lattice of right annihilator ideals of R as [math]\displaystyle{ \mathcal{RA}\, }[/math]. It is known that [math]\displaystyle{ \mathcal{LA}\, }[/math] satisfies the A.C.C. if and only if [math]\displaystyle{ \mathcal{RA}\, }[/math] satisfies the D.C.C., and symmetrically [math]\displaystyle{ \mathcal{RA}\, }[/math] satisfies the A.C.C. if and only if [math]\displaystyle{ \mathcal{LA}\, }[/math] satisfies the D.C.C. If either lattice has either of these chain conditions, then R has no infinite orthogonal sets of idempotents. [7][8]
If R is a ring for which [math]\displaystyle{ \mathcal{LA}\, }[/math] 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
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 M → M along the Hom-tensor adjunction.
More generally, given a bilinear map of modules [math]\displaystyle{ F\colon M \times N \to P }[/math], the annihilator of a subset [math]\displaystyle{ S \subseteq M }[/math] is the set of all elements in [math]\displaystyle{ N }[/math] that annihilate [math]\displaystyle{ S }[/math]:
- [math]\displaystyle{ \operatorname{Ann}(S) := \{ n \in N \mid \forall s \in S: F(s,n) = 0 \} . }[/math]
Conversely, given [math]\displaystyle{ T \subseteq N }[/math], one can define an annihilator as a subset of [math]\displaystyle{ M }[/math].
The annihilator gives a Galois connection between subsets of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math], and the associated closure operator is stronger than the span. In particular:
- annihilators are submodules
- [math]\displaystyle{ \operatorname{Span}S \leq \operatorname{Ann}(\operatorname{Ann}(S)) }[/math]
- [math]\displaystyle{ \operatorname{Ann}(\operatorname{Ann}(\operatorname{Ann}(S))) = \operatorname{Ann}(S) }[/math]
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 [math]\displaystyle{ V \times V \to K }[/math] 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.
- Annihilators are used to define left Rickart rings and Baer rings.
- The set of (left) zero divisors DS of S can be written as
- [math]\displaystyle{ D_S = \bigcup_{x \in S \setminus \{0\}}{\mathrm{Ann}_R(x)}. }[/math]
- (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.
- When R is commutative and Noetherian, the set [math]\displaystyle{ D_R }[/math] is precisely equal to the union of the associated primes of the R-module R.
See also
Notes
- ↑ 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". https://stacks.math.columbia.edu/tag/00L2.
- ↑ "Lemma 10.39.9 (00L3)—The Stacks project". https://stacks.math.columbia.edu/tag/00L3.
- ↑ "Lemma 10.39.9 (00L3)—The Stacks project". https://stacks.math.columbia.edu/tag/00L3.
- ↑ Anderson & Fuller 1992, p. 322.
- ↑ 8.0 8.1 Lam 1999.
References
- Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3
- 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, 189, Berlin, New York: Springer-Verlag, pp. 228–232, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5
- Richard S. Pierce. Associative algebras. Graduate Texts in Mathematics, Vol. 88, Springer-Verlag, 1982, ISBN:978-0-387-90693-5
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