Annihilator (ring theory)

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

Definitions[edit]

Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M. The annihilator, denoted AnnR(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 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 \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 AnnR(M) = 0, then M is called a faithful module.

Properties[edit]

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 AnnR(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 AnnR(N) is a subset of AnnR(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/AnnR(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/AnnR(M)-module, M is automatically a faithful module.

Chain conditions on annihilator ideals[edit]

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[edit]

When R is commutative and M is an R-module, we may describe AnnR(M) as the kernel of the action map REndR(M) determined by the adjunct map of the identity MM 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[edit]

D_S = \bigcup_{x \in S,\, x \neq 0}{\mathrm{Ann}_R\,(x)}.

(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, the set DR is precisely equal to the union of the minimal prime ideals of R.

See also[edit]

Notes[edit]

  1. ^ Pierce (1982), p. 23.
  2. ^ Pierce (1982), p. 23, Lemma b, item (i).

References[edit]