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 words connection to and/or connection:
“One must always maintain one’s connection to the past and yet ceaselessly pull away from it. To remain in touch with the past requires a love of memory. To remain in touch with the past requires a constant imaginative effort.”
—Gaston Bachelard (1884–1962)
“It may comfort you to know that if your child reaches the age of eleven or twelve and you have a good bond or relationship, no matter how dramatic adolescence becomes, you children will probably turn out all right and want some form of connection to you in adulthood.”
—Charlotte Davis Kasl (20th century)