Definition:Annihilator on Algebraic Dual

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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.