Definition:Annihilator of Ideal of Ring

Definition

Let $A$ be a commutative ring with unity.

Let $I \subseteq A$ be an ideal.

Definition 1

The annihilator of $I$ is the ideal consisting of the elements $a \in A$ such that $a \cdot x = 0$ for all $x \in I$, where $0 \in A$ is its zero.

Definition 2

The annihilator of $I$ is the ideal quotient $\ideal {0 : I}$, where $0$ is the zero ideal.