Prime ideal

In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. Prime ideals have a simpler description for commutative rings, so we consider this case separately below. This article only covers ideals of ring theory. Prime ideals in order theory are treated in the article on ideals in order theory.

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If R is a commutative ring, then an ideal P of R is prime if it has the following two properties:

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• whenever a, b are two elements of R such that their product ab lies in 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

Prime ideal: Prime ideals for commutative rings ►



 

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