Definition:Annihilator on Algebraic Dual

Definition

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:

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

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

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.

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