Prime ideal

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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 Y2X3X − 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 set-theoretic 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  \mathbb{C}[x,y] with prime ideals P = (x2 + y2 − 1) and Q = (x) (the ideals generated by x2 + y2 − 1 and x respectively). Their sum P + Q = (x2 + y2 − 1 , x) = (y2 − 1 , x) however is not prime: y2 − 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 non-zero element of R and the ideal (q2) 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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