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:
“The prime lesson the social sciences can learn from the natural sciences is just this: that it is necessary to press on to find the positive conditions under which desired events take place, and that these can be just as scientifically investigated as can instances of negative correlation. This problem is beyond relativity.”
—Ruth Benedict (1887–1948)
“Institutional psychiatry is a continuation of the Inquisition. All that has really changed is the vocabulary and the social style. The vocabulary conforms to the intellectual expectations of our age: it is a pseudo-medical jargon that parodies the concepts of science. The social style conforms to the political expectations of our age: it is a pseudo-liberal social movement that parodies the ideals of freedom and rationality.”
—Thomas Szasz (b. 1920)
“The next time the novelist rings the bell I will not stir though the meeting-house burn down.”
—Henry David Thoreau (1817–1862)