# Definition:Annihilator on Algebraic Dual

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## Definition

Let $R$ be a commutative ring with unity.

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 $M$** is denoted and defined as:

- $M^\circ := \set {t \in G^*: \forall x \in M: \map t x = 0}$

## Also denoted as

Some sources denote this as $\map {\operatorname {Ann} } M$.

## 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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations