Prime ideal
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.
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
Examples
If R denotes the ring C[X, Y] 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 Z[X] 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.