# Definition:Annihilator

## Contents

## 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:

- $\ M^\circ := \left\{{t' \in G^*: \forall x \in M: t' \left({x}\right) = 0}\right\}$

### 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:

- $B: R \times \Z: \left({r, n}\right) \mapsto n \cdot r$

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

- $n \cdot r = r + r + \cdots \left({n}\right) \cdots r$

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

- $B \left({r, n}\right) = n \cdot r$

Let $D \subseteq R$.

Then the **annihilator** of $D$ is defined as:

- $\operatorname{Ann} \left({D}\right) = \left\{{n \in \Z: \forall d \in D: n \cdot d = 0_R}\right\}$

or, when $D = R$:

- $\operatorname{Ann} \left({R}\right) = \left\{{n \in \Z: \forall r \in R: n \cdot r = 0_R}\right\}$

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**.