Prime Ideals For Commutative Rings
An ideal P of a commutative ring R is prime if it has the following two properties:
- If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P,
- P is not equal to the whole ring R.
This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say
- A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z.
Read more about this topic: Prime Ideal
Famous quotes containing the words prime, ideals and/or rings:
“I did not know I was in my prime until afterwards.”
—Mason Cooley (b. 1927)
“It does not follow, because our difficulties are stupendous, because there are some souls timorous enough to doubt the validity and effectiveness of our ideals and our system, that we must turn to a state controlled or state directed social or economic system in order to cure our troubles.”
—Herbert Hoover (1874–1964)
“It is told that some divorcees, elated by their freedom, pause on leaving the courthouse to kiss a front pillar, or even walk to the Truckee to hurl their wedding rings into the river; but boys who recover the rings declare they are of the dime-store variety, and accuse the throwers of fraudulent practices.”
—Administration in the State of Neva, U.S. public relief program. Nevada: A Guide to the Silver State (The WPA Guide to Nevada)