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).
Read more about this topic: Prime Ideal
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