Annihilator (ring theory)

In mathematics, specifically module theory, annihilators are a concept that formalizes torsionand 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_"R""S" 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 $Fcolon\; M\; imes\; N\; o\; P$, 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 $V\; imes\; V\; o\; K$ is called the orthogonal complement.

Properties

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" 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") = "c"0 = 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_"R""M"-module.

Relations to other properties of rings

*The set of (left) zero divisors "D"_"S" of "S" can be written as:::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