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:
“Ay, look: high heaven and earth ail from the prime foundation;
All thoughts to rive the heart are here, and all are vain:
Horror and scorn and hate and fear and indignation—
Oh, why did I awake? When shall I sleep again?”
—A.E. (Alfred Edward)
“The measure discriminates definitely against products which make up what has been universally considered a program of safe farming. The bill upholds as ideals of American farming the men who grow cotton, corn, rice, swine, tobacco, or wheat and nothing else. These are to be given special favors at the expense of the farmer who has toiled for years to build up a constructive farming enterprise to include a variety of crops and livestock.”
—Calvin Coolidge (1872–1933)
“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)