methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.

Examples and properties

* The definition can be rephrased in a more apparently symmetrical manner: a proper ideal is primary if, whenever , or are elements of , or both and lie in , the radical of ; i.e.,
* A proper ideal of is primary if and only if every

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