Editing Prime ideal (section)

{{Short description|Ideal in a ring which has properties similar to prime elements}}
{{about|ideals in [[ring theory]]|prime ideals in order theory|Ideal (order theory)#Prime ideals}}
{{distinguish|text=[[Primary ideal]]}}
[[File:A portion of the lattice of ideals of Z illustrating prime, semiprime and primary ideals SVG.svg|thumb|right|upright=1.3|A [[Hasse diagram]] of a portion of the [[lattice (order)|lattice]] of [[ideal (ring theory)|ideals]] of the integers <math>\Z.</math> The purple nodes indicate prime ideals. The purple and green nodes are [[semiprime ideal]]s, and the purple and blue nodes are [[primary ideal]]s.]] In [[algebra]], a '''prime ideal''' is a [[subset]] of a [[ring (mathematics)|ring]] that shares many important properties of a [[prime number]] in the ring of [[Integer#Algebraic properties|integers]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | author-link=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> The prime ideals for the integers are the sets that contain all the [[multiple (mathematics)|multiples]] of a given prime number, together with the [[zero ideal]].

[[Primitive ideal]]s are prime, and prime ideals are both [[primary ideal|primary]] and [[semiprime ideal|semiprime]].