Annihilator (ring theory)

In mathematics, specifically module theory, annihilators are a concept that formalizes torsion and generalizes torsion and orthogonal complement.

Definition

Let R be a ring, and let M be a left R-module. Choose a subset S of M. The annihilator Ann_RS of S is the set of all elements r in R such that for each s in S, rs = 0: it is the set of all elements that annihilate S (the elements for which S is torsion).

More generally, given a bilinear map of modules $Fcolon\; M\; times\; N\; to\; P$, 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.

Properties

The annihilator of a single element x is usually written Ann_Rx instead of Ann_R{x}. If the ring R can be understood from the context, the subscript R is usually omitted.

Annihilators are always one-sided ideals of their ring: If a and b both annihilate S, then for each s in S, (a + b)s = as + bs = 0, and for any c in R, (ca)s = c(as) = c0 = 0. The annihilator of M is even a two-sided ideal: (ac)s = a(cs) = 0, since cs is another element of M.

M is always a faithful R/Ann_RM-module.

Relations to other properties of rings

• The set of (left) zero divisors D_S of S can be written as

$D\_S\; =\; bigcup\_\{x\; in\; S,,\; x\; neq\; 0\}\{mathrm\{Ann\}\_R,x\}.$

In particular D is the set of (left) zero divisors of R when S = R and R acts on itself as a left R-module.

References