Definition:Annihilator of Ideal of Ring
Jump to navigation
Jump to search
This page has been identified as a candidate for refactoring. In particular:
multiple definitions
multiple definitions
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
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.
Also see