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 of a commutative ring is prime if it has the following two properties: If and are two elements of such that their product is an element of , then is in or is in , is not the whole ring . This generalizes the following property of prime numbers, known as Euclid's lemma: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say A positive integer is a prime number if and only if is a prime ideal in Examples 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 denotes the ring of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial is a prime ideal (see elliptic curve). In the ring of all polynomials with integer coefficients, the ideal generated by and is a prime ideal. It consists of all those polynomials whose constant coefficient is even. In any ring , a maximal ideal is an ideal that is maximal in the set of all proper ideals of , i.e. is contained in exactly two ideals of , namely itself and the whole ring . 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. For the UFD Hilbert's Nullstellensatz states that every maximal ideal is of the form If is a smooth manifold, is the ring of smooth real functions on , and is a point in , then the set of all smooth functions with forms a prime ideal (even a maximal ideal) in . Non-examples Consider the