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 purpose of being four is to enjoy being four—of secondary importance is to prepare for being five.”
—Jim Trelease (20th century)
“But I would emphasize again that social and economic solutions, as such, will not avail to satisfy the aspirations of the people unless they conform with the traditions of our race, deeply grooved in their sentiments through a century and a half of struggle for ideals of life that are rooted in religion and fed from purely spiritual springs.”
—Herbert Hoover (1874–1964)
“We will have rings and things, and fine array,
And kiss me, Kate, we will be married o’ Sunday.”
—William Shakespeare (1564–1616)