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. Prime ideals have a simpler description for commutative rings, so we consider this case separately below. This article only covers ideals of ring theory. Prime ideals in order theory are treated in the article on ideals in order theory.
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Prime ideals 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 a contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime; the converse 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 maximal ideal (see above) is prime; an ideal I in the commutative ring R is a maximal ideal if and only if the factor ring R/I is a field.
- Every nonzero commutative ring contains at least one prime ideal. In fact, it contains at least one maximal ideal, which can be proven using Zorn's lemma.
- A commutative ring is an integral domain if and only if {0} is a prime ideal.
- A commutative 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.
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.
- Any primitive ideal is prime.
- The zero ideal of any prime ring is prime.
Properties
- An ideal P is prime if and only if for two ideals A and B, AB = P implies that either A or B is contained in P. This is sometimes taken as the definition of a prime ideal. This is close to the historical point of view of ideals as ideal numbers, as "A is contained in P" is another way of saying "P divides A".