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:
“The Transcendentalist adopts the whole connection of spiritual doctrine. He believes in miracle, in the perpetual openness of the human mind to new influx of light and power; he believes in inspiration, and in ecstacy.”
—Ralph Waldo Emerson (1803–1882)