# 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 on Algebraic Dual

Let $R$ be a commutative ring.

Let $G$ be a module over $R$.

Let $G^*$ be the algebraic dual of $G$.

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

The annihilator of $M$, denoted $M^\circ$, is defined as:

$\ M^\circ := \set {t' \in G^*: \forall x \in M: \map {t'} x = 0}$

### Annihilator as Integral Multiples of Ring Elements

A common instance is when $N$ is the ring of integers $\Z$ and $M$ is a ring or a field $\left({R, +, \times}\right)$:

Let $B: R \times \Z$ be a bilinear mapping defined as:

$B: R \times \Z: \tuple {r, n} \mapsto n \cdot r$

where $n \cdot r$ defined as an integral multiple of $r$:

$n \cdot r = r + r + \cdots \paren n \cdots r$

Note the change of order of $r$ and $n$:

$\map B {r, n} = n \cdot r$

Let $D \subseteq R$ be a subring of $R$.

Then the annihilator of $D$ is defined as:

$\map {\operatorname {Ann} } D = \set {n \in \Z: \forall d \in D: n \cdot d = 0_R}$

or, when $D = R$:

$\map {\operatorname {Ann} } R = \set {n \in \Z: \forall r \in R: n \cdot r = 0_R}$

It is seen to be, therefore, the set of all integers whose integral multiples, with respect to the elements of a ring or a field, are all equal to the zero of that ring or field.

### Trivial Annihilator

From Annihilator of Ring Always Contains Zero, we have that $0 \in \operatorname{Ann} \left({R}\right)$ whatever the ring $R$ is.

$R$ is said to have a trivial annihilator if and only if its annihilator $\operatorname{Ann} \left({R}\right)$ consists only of the integer $0$.

## 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.

The pronunciation of annihilator is something like an-nile-a-tor.