Definition:Annihilator

Definition
Let $R$ be a commutative ring.

Let $M$ and $N$ be modules over $R$.

Let $B : M \times N \to R$ be a bilinear mapping.

The annihilator of $D \subseteq M$, denoted $\operatorname{Ann}_N \left({D}\right)$ is the set:


 * $\left\{{n \in N : \forall d \in D: B \left({d, n}\right) = 0}\right\}$

Special Cases
Various definitions of the annihilator can be found in the literature, including:

Annihilator as Linear Forms
Let $N = M^*$ be the algebraic dual of $M$.

Let $B : M \times G \to R : \left({m, n}\right) \mapsto n \left({m}\right)$.

Let $L$ be a submodule of $M$.

In this case the annihilator of $L$ is:


 * $L^\circ := \left\{{n \in M^*: \forall \ell \in L: n \left({\ell}\right) = 0}\right\}$

Annihilator as Ring Elements
Let $N = R$.

Let $B \left({m, r}\right) = r \cdot m$, where $\cdot$ is the multiplication from the module structure.

In this case, for $D \subseteq M$:


 * $\operatorname{Ann}_R \left({D}\right) = \left\{{r \in R : \forall d \in D: r \cdot d = 0}\right\}$

Annihilator as Integral Multiples of Ring Elements
The usual case is when $N$ is the ring of integers $\Z$ and $M$ is a ring or a field $\left({R, +, \times}\right)$:

Linguistic Note
The word annihilator calls to mind a force of destruction which removes something from existence.

In fact, the word is a compound construct based on the Latin nihil, which means nothing.

Thus annihilator can be seen to mean, literally, an entity which causes (something) to become nothing.