# Definition:Annihilator of Ring

## Definition

Let $\struct {R, +, \times}$ be a ring or, more usually, a field.

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 $0$.

## Also see

## 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**.

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.2$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**annihilator**:**3.**