Editing Prime ideal (section)

===Examples===
* Any [[primitive ideal]] is prime.
* As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals.
* A ring is a [[prime ring]] if and only if the zero ideal is a prime ideal, and moreover a ring is a [[integral domain|domain]] if and only if the zero ideal is a completely prime ideal.
* Another fact from commutative theory echoed in noncommutative theory is that if {{mvar|A}} is a nonzero {{mvar|R}}-[[module (mathematics)|module]], and {{mvar|P}} is a maximal element in the [[poset]] of [[Annihilator (ring theory)|annihilator]] ideals of submodules of {{mvar|A}}, then {{mvar|P}} is prime.