Definition:Annihilator on Algebraic Dual

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 see

• Definition:Annihilator

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
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): annihilator (vector spaces)