Definition:Annihilator of Ideal of Ring
multiple definitions
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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 $(0 : I)$, where $0$ is the zero ideal.
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