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In algebra (which is a branch of mathematics), a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers.^{[1]}^{[2]} The prime ideals for the integers are the sets that contain all the multiples of a given prime number.
A primary ideal is a generalization of a prime ideal.
Contents
Prime ideals for commutative rings
An ideal P of a commutative ring R is prime if it has the following two properties:
 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
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.
 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 2 ideals of R, namely M itself and the entire 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.
 If M is a smooth manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set of all smooth functions f with f(x) = 0 forms a prime ideal (even a maximal ideal) in R.
Properties
 An ideal I in the ring R is prime if and only if the factor ring R/I is an integral domain. In particular, a commutative ring is an integral domain if and only if {0} is a prime ideal.
 An ideal I is prime if and only if its settheoretic complement is multiplicatively closed.
 Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal) which is a direct consequence of Krull's theorem.
 The set of all prime ideals (the spectrum of a ring) contains minimal elements (called minimal prime). Geometrically, these correspond to irreducible components of the spectrum.
 The preimage of a prime ideal under a ring homomorphism is a prime ideal.
 The sum of two prime ideals is not necessarily prime. For an example, consider the ring with prime ideals P = (x^{2} + y^{2} − 1) and Q = (x) (the ideals generated by x^{2} + y^{2} − 1 and x respectively). Their sum P + Q = (x^{2} + y^{2} − 1 , x) = (y^{2} − 1 , x) however is not prime: y^{2} − 1 = (y − 1) (y + 1) is in P + Q but its two factors are not. Alternatively, note that the quotient ring has zero divisors so it is not an integral domain and thus P + Q cannot be prime.
 If every proper ideal in a commutative ring R with at least two elements is prime, then the ring is a field. (If the ideal (0) is prime, then the ring R is an integral domain. If q is any nonzero element of R and the ideal (q^{2}) is prime, then it contains q and then q is invertible.)
 A nonzero principal ideal is prime if and only if it is generated by a prime element. In a UFD, every nonzero prime ideal contains a prime element.
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