Annihilator (ring theory)

In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality.

Definitions

Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M. The annihilator, denoted Ann_R(S), of S is the set of all elements r in R such that for each s in S, rs = 0:^[1] In set notation,

\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 torsion). 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 $\ell.\mathrm{Ann}_R(S)\,$ and $r.\mathrm{Ann}_R(S)\,$ 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. The proof is straightforward: 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. (A similar proof follows for subsets of right modules to show that the annihilator is a right ideal.)

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.^[2]

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 it is easy to check that equality holds.

M may be also viewed as a R/Ann_R(M)-module using the action $\overline{r}m:=rm\,$ . 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.

Chain conditions on annihilator ideals

The lattice of ideals of the form $\ell.\mathrm{Ann}_R(S)\,$ 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 $\mathcal{LA}\,$ and the lattice of right annihilator ideals of R as $\mathcal{RA}\,$ . It is known that $\mathcal{LA}\,$ satisfies the A.C.C. if and only if $\mathcal{RA}\,$ satisfies the D.C.C., and symmetrically $\mathcal{RA}\,$ satisfies the A.C.C. if and only if $\mathcal{LA}\,$ 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) (Lam 1999)

If R is a ring for which $\mathcal{LA}\,$ 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 $F\colon M \times N \to P$ , the annihilator of a subset $S \subset M$ is the set of all elements in $N$ that annihilate $S$ :

\mbox{Ann}\,(S) := \{ n \in N \mid \forall s \in S, F(s,n) = 0\}

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

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

annihilators are submodules
$\mbox{Span}\,(S) \leq \mbox{Ann}(\mbox{Ann}\,(S))$
$\mbox{Ann}(\mbox{Ann}(\mbox{Ann}\,(S))) = \mbox{Ann}\,(S)$

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 $V \times V \to K$ is called the orthogonal complement.

Relations to other properties of rings

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

D_S = \bigcup_{x \in S,\, x \neq 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 $D_R$ is precisely equal to the union of the associated prime ideals of R.

Notes

↑ Pierce (1982), p. 23.
↑ Pierce (1982), p. 23, Lemma b, item (i).

References

Weisstein, Eric W., "Annihilator", MathWorld.
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, 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, 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