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 ${\displaystyle M}$ over a ring ${\displaystyle R}$ is the set of elements in ${\displaystyle R}$ that always act as multiplication by ${\displaystyle 0}$ on ${\displaystyle M}$. The prototypical example for an annihilator over a commutative ring can be understood by taking the quotient ring ${\displaystyle R/I}$ and considering it as a ${\displaystyle R}$-module. Then, the annihilator of ${\displaystyle R/I}$ is the ideal ${\displaystyle I}$ since all of the ${\displaystyle i\in I}$ act via the zero map on ${\displaystyle R/I}$. This shows how the ideal ${\displaystyle I}$ can be thought of as the set of torsion elements in the base ring ${\displaystyle R}$ for the module ${\displaystyle R/I}$. Also, notice that any element ${\displaystyle r\in R}$ that isn't in ${\displaystyle I}$ will have a non-zero action on the module ${\displaystyle R/I}$, implying the set ${\displaystyle R-I}$ can be thought of as the set of orthogonal elements to the ideal ${\displaystyle I}$.

For noncommutative rings ${\displaystyle R}$, there is a similar notion of the annihilator for left and right modules, called the left-annihilator and the right-annihilator.

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, rs = 0.[1] In set notation,

${\displaystyle \mathrm {Ann} _{R}(S)=\{r\in R\mid \forall s\in S:\,rs=0\}}$

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 R module, the notation must be modified slightly to indicate the left or right side. Usually ${\displaystyle \ell .\mathrm {Ann} _{R}(S)\,}$ and ${\displaystyle r.\mathrm {Ann} _{R}(S)\,}$ or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.

If M is an R-module and Ann_R(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 Ann_R(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 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 ${\displaystyle {\overline {r}}m:=rm\,}$. 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 ${\displaystyle R}$ be a commutative ring and ${\displaystyle M}$ a finite ${\displaystyle R}$-module.

Relation to support

Recall that the support of a module is defined as

${\displaystyle {\text{Supp}}(M)=\{{\mathfrak {p}}\in {\text{Spec}}(R)|M_{\mathfrak {p}}\neq 0\}}$

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

${\displaystyle V({\text{Ann}}_{R}(M))={\text{Supp}}(M)}$

where ${\displaystyle V(\cdot )}$ is the set of prime ideals containing the subset[4].

Short exact sequences

Given a short exact sequence of modules

${\displaystyle 0\to M'\to M\to M''\to 0}$

the support property

${\displaystyle {\text{Supp}}(M)={\text{Supp}}(M')\cup {\text{Supp}}(M'')}$[5]

together with the relation with the annihilator implies

${\displaystyle V({\text{Ann}}_{R}(M))=V({\text{Ann}}_{R}(M'))\cup V({\text{Ann}}_{R}(M''))}$

Hence

${\displaystyle {\text{Ann}}_{R}(M)={\text{Ann}}_{R}(M')\cap {\text{Ann}}_{R}(M'')}$

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

${\displaystyle {\text{Ann}}_{R}\left(\bigoplus _{i=1}^{n}M_{i}\right)=\bigcap _{i=1}^{n}{\text{Ann}}_{R}(M_{i})}$

Quotient modules and annihilators

Given an ideal ${\displaystyle I\subseteq R}$ and let ${\displaystyle M}$ be a finite module, then there is the relation

${\displaystyle {\text{Supp}}(M/IM)={\text{Supp}}(M)\cap V(I)}$

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

${\displaystyle V({\text{Ann}}_{R}(M/IM))=V({\text{Ann}}_{R}(M))\cap V(I)}$

Annihilator of quotient ring

In particular, if ${\displaystyle M=R}$ then the annihilator of ${\displaystyle R/I}$ can be found explicitly using

${\displaystyle {\begin{aligned}V({\text{Ann}}_{R}(R/I))&=V((0))\cap V(I)\\&=V(I)\end{aligned}}}$

Hence the annihilator of ${\displaystyle R/I}$ is just ${\displaystyle I}$.

Examples

Over the integers

Over ${\displaystyle \mathbb {Z} }$ 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

${\displaystyle {\text{Ann}}_{\mathbb {Z} }(\mathbb {Z} ^{\oplus k})=\{0\}=(0)}$

since the only element killing each of the ${\displaystyle \mathbb {Z} }$ is ${\displaystyle 0}$. For example, the annihilator of ${\displaystyle \mathbb {Z} /2\oplus \mathbb {Z} /3}$ is

${\displaystyle {\text{Ann}}_{\mathbb {Z} }(\mathbb {Z} /2\oplus \mathbb {Z} /3)=(6)=({\text{lcm}}(2,3))}$

the ideal generated by ${\displaystyle (6)}$. In fact the annihilator of a torsion module

${\displaystyle M\cong \bigoplus _{i=1}^{n}(\mathbb {Z} /a_{i})^{\oplus k_{i}}}$

is isomorphic to the ideal generated by their least common multiple, ${\displaystyle ({\text{lcm}}(a_{1},\ldots ,a_{n}))}$. 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 ${\displaystyle R}$. Recall that the definition of finiteness of ${\displaystyle M}$ implies there exists a right-exact sequence, called a presentation, given by

${\displaystyle R^{\oplus l}\xrightarrow {\phi } R^{\oplus k}\to M\to 0}$

where ${\displaystyle \phi }$ is in ${\displaystyle {\text{Mat}}_{k,l}(R)}$. Writing ${\displaystyle \phi }$ explicitly as a matrix gives it as

${\displaystyle \phi ={\begin{bmatrix}\phi _{1,1}&\cdots &\phi _{1,n}\\\vdots &&\vdots \\\phi _{n,1}&\cdots &\phi _{n,n}\end{bmatrix}}}$

hence ${\displaystyle M}$ has the direct sum decomposition

${\displaystyle M=\bigoplus _{i=1}^{k}{\frac {R}{(\phi _{i,1}(1),\ldots ,\phi _{i,n}(1))}}}$

If we write each of these ideals as

${\displaystyle I_{i}=(\phi _{i,1}(1),\ldots ,\phi _{i,n}(1))}$

then the ideal ${\displaystyle I}$ given by

${\displaystyle V(I)=\bigcup _{i=1}^{n}V(I_{i})}$

presents the annihilator.

Over k[x,y]

Over the commutative ring ${\displaystyle k[x,y]}$ for a field ${\displaystyle k}$, the annihilator of the module

${\displaystyle M={\frac {k[x,y]}{(x^{2}-y)}}\oplus {\frac {k[x,y]}{(y-3)}}}$

is given by the ideal

${\displaystyle {\text{Ann}}_{k[x,y]}(M)=((x^{2}-y)(y-3))}$

Chain conditions on annihilator ideals

The lattice of ideals of the form ${\displaystyle \ell .\mathrm {Ann} _{R}(S)\,}$ 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 ${\displaystyle {\mathcal {LA}}\,}$ and the lattice of right annihilator ideals of R as ${\displaystyle {\mathcal {RA}}\,}$. It is known that ${\displaystyle {\mathcal {LA}}\,}$ satisfies the A.C.C. if and only if ${\displaystyle {\mathcal {RA}}\,}$ satisfies the D.C.C., and symmetrically ${\displaystyle {\mathcal {RA}}\,}$ satisfies the A.C.C. if and only if ${\displaystyle {\mathcal {LA}}\,}$ 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) harv error: no target: CITEREFAnderson1992,_p.322 (help) (Lam 1999)

If R is a ring for which ${\displaystyle {\mathcal {LA}}\,}$ 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 rings

When R is commutative and M is an R-module, 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 Hom-tensor adjunction.

More generally, given a bilinear map of modules ${\displaystyle F\colon M\times N\to P}$, the annihilator of a subset ${\displaystyle S\subseteq M}$ is the set of all elements in ${\displaystyle N}$ that annihilate ${\displaystyle S}$:

${\displaystyle \operatorname {Ann} (S):=\{n\in N\mid \forall s\in S:F(s,n)=0\}.}$

Conversely, given ${\displaystyle T\subseteq N}$, one can define an annihilator as a subset of ${\displaystyle M}$.

The annihilator gives a Galois connection between subsets of ${\displaystyle M}$ and ${\displaystyle N}$, and the associated closure operator is stronger than the span. In particular:

• annihilators are submodules
• ${\displaystyle \operatorname {Span} (S)\leq \operatorname {Ann} (\operatorname {Ann} (S))}$
• ${\displaystyle \operatorname {Ann} (\operatorname {Ann} (\operatorname {Ann} (S)))=\operatorname {Ann} (S)}$

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 ${\displaystyle V\times V\to K}$ 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 D_S of S can be written as

${\displaystyle D_{S}=\bigcup _{x\in S\setminus \{0\}}{\mathrm {Ann} _{R}\,(x)}.}$

(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 R-module.

• When R is commutative and Noetherian, the set ${\displaystyle D_{R}}$ is precisely equal to the union of the associated primes of the R-module R.

See also

• Socle
• Support of a module

Notes

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.

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, 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, Berlin, New York: Springer-Verlag, pp. 228–232, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5, MR 1653294
• Richard S. Pierce. Associative algebras. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN 978-0-387-90693-5