Editing Prime ideal (section)

==Connection to maximality==
Prime ideals can frequently be produced as maximal elements of certain collections of ideals. For example:
* An ideal maximal with respect to having empty intersection with a fixed m-system is prime.
* An ideal maximal among [[Annihilator (ring theory)|annihilators]] of submodules of a fixed {{mvar|R}}-module {{mvar|M}} is prime.
* In a commutative ring, an ideal maximal with respect to being non-principal is prime.<ref>Kaplansky ''Commutative rings'', p. 10, Ex 10.</ref>
* In a commutative ring, an ideal maximal with respect to being not countably generated is prime.<ref>Kaplansky ''Commutative rings'', p. 10, Ex 11.</ref>