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 should always remember that the work of art is invariably the creation of a new world, so that the first thing we should do is to study that new world as closely as possible, approaching it as something brand new, having no obvious connection with the worlds we already know. When this new world has been closely studied, then and only then let us examine its links with other worlds, other branches of knowledge.”
—Vladimir Nabokov (1899–1977)