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:
“Sometimes it takes years to really grasp what has happened to your life. What do you do after you are world-famous and nineteen or twenty and you have sat with prime ministers, kings and queens, the Pope? What do you do after that? Do you go back home and take a job? What do you do to keep your sanity? You come back to the real world.”
—Wilma Rudolph (1940–1994)
“My own ideals for the university are those of a genuine democracy and serious scholarship. These two, indeed, seem to go together.”
—Woodrow Wilson (1856–1924)
“The next time the novelist rings the bell I will not stir though the meeting-house burn down.”
—Henry David Thoreau (1817–1862)