Prime Ideal - Connection To Maximality

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 annihilators of submodules of a fixed R module M is prime.
• In a commutative ring, an ideal maximal with respect to being non-principal is prime.
• In a commutative ring, an ideal maximal with respect to being not countably generated is prime.

This unusual affinity has been studied further in (Lam & Reyes 2008).