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Concept
Prime ideal
Summary
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.
Primitive ideals are prime, and prime ideals are both primary and semiprime.
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 the whole ring R.
This generalizes the following property of prime numbers, known as Euclid's lemma: 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 is a prime ideal in
A simple example: In the ring the subset of even numbers is a prime ideal.
Given an integral domain , any prime element generates a principal prime ideal . Eisenstein's criterion for integral domains (hence UFDs) is an effective tool for determining whether or not an element in a polynomial ring is irreducible. For example, take an irreducible polynomial in a polynomial ring over some field .
If R denotes the ring of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y 2 − X 3 − X − 1 is a prime ideal (see elliptic curve).
In the ring of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. It consists of all those polynomials whose constant coefficient is even.
In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is contained in exactly two ideals of R, namely M itself and the whole ring R. Every maximal ideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general.
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