Prime ideal

In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. This article only covers ideals of ring theory. Prime ideals in order theory are treated in the article on ideals in order theory.

A primary ideal is a generalization of a prime ideal.

Formal definition

• An ideal ${\mathfrak {p}}$ of a ring $R$ is prime if and only if it is a proper ideal (ie ${\mathfrak {p}}\neq R$) and for any two ideals ${\mathfrak {a}}$ and ${\mathfrak {b}}$ in $R$, if ${\mathfrak {ab}}\subseteq {\mathfrak {p}}$, then ${\mathfrak {a}}\subseteq {\mathfrak {p}}$ or ${\mathfrak {b}}\subseteq {\mathfrak {p}}$.

This is close to the historical point of view of ideals as ideal numbers, as for the ring Z "A is contained in P" is another way of saying "P divides A", and the unit ideal R represents unity.

Prime ideals for commutative rings

Prime ideals have a simpler description for commutative rings: if R is a commutative ring, then an ideal P of 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

A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z.

Examples

• If R denotes the ring C[X, Y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y² − X³ − 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 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 commutative ring R is prime if and only if the factor ring R/I is an integral domain.
• Every nonzero commutative unital 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.
• A commutative unital ring is an integral domain if and only if {0} is a prime ideal.
• A commutative unital ring is a field if and only if {0} is its only prime ideal, or equivalently, if and only if {0} is a maximal ideal.
• 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. To see this note the quotient ring has zero divisors implying that the quotient is not an integral domain and thus P + Q cannot be prime.

Uses

One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach, one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into a topological space and can thus define generalizations of varieties called schemes, which find applications not only in geometry, but also in number theory.

The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the important property of unique factorisation expressed in the fundamental theorem of arithmetic does not hold in every ring of algebraic integers, but a substitute was found when Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.

Prime ideals for noncommutative rings

If R is a noncommutative ring, then an ideal P of R is prime if it has the following two properties:

• whenever a, b are two elements of R such that for all elements r of R, their product arb lies in P, then a is in P or b is in P.
• P is not equal to the whole ring R.

For commutative rings this definition is equivalent to the one given in the previous section. For noncommutative rings, the two definitions are different. An ideal such that ab in P implies that a or b is in P is called a completely prime ideal. Completely prime ideals are prime ideals, but the converse is not true. For example, the zero ideal in the ring of n × n matrices is a prime ideal, but it is not completely prime.

Examples

• Any maximal ideal is prime (in a commutative ring with unity).

• Any primitive ideal is prime.

• The zero ideal of any prime ring is prime.