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)
“With the breakdown of the traditional institutions which convey values, more of the burdens and responsibility for transmitting values fall upon parental shoulders, and it is getting harder all the time both to embody the virtues we hope to teach our children and to find for ourselves the ideals and values that will give our own lives purpose and direction.”
—Neil Kurshan (20th century)
“You held my hand
and were instant to explain
the three rings of danger.”
—Anne Sexton (1928–1974)