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:
“One wants in a Prime Minister a good many things, but not very great things. He should be clever but need not be a genius; he should be conscientious but by no means strait-laced; he should be cautious but never timid, bold but never venturesome; he should have a good digestion, genial manners, and, above all, a thick skin.”
—Anthony Trollope (1815–1882)
“The real weakness of England lies, not in incomplete armaments or unfortified coasts, not in the poverty that creeps through sunless lanes, or the drunkenness that brawls in loathsome courts, but simply in the fact that her ideals are emotional and not intellectual.”
—Oscar Wilde (1854–1900)
“We will have rings and things, and fine array,
And kiss me, Kate, we will be married o’ Sunday.”
—William Shakespeare (1564–1616)