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
Famous quotes containing the word connection:
“We will have to give up the hope that, if we try hard, we somehow will always do right by our children. The connection is imperfect. We will sometimes do wrong.”
—Judith Viorst (20th century)