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

Definition

An ideal P of a commutative ring R is prime if it has the following two properties: ab is an element of P, then a is in P or b is in P, This generalizes the following property of prime numbers, known as Euclid's lemma: 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

Y 2 − X 3 − X − 1 is a prime ideal (see elliptic curve). 2 and X is a prime ideal. The ideal consists of all polynomials constructed by taking 2 times an element of \Z[X] and adding it to X times another polynomial in \Z[X] (which converts the constant coefficient in the latter polynomial into a linear coefficient). Therefore, the resultant ideal consists of all those polynomials whose constant coefficient is even. f (x) = 0 forms a prime ideal (even a maximal ideal) in R.

Non-examples

Properties

I in the ring R (with unity) is prime if and only if the factor ring R/I is an integral domain. In particular, a commutative ring (with unity) is an integral domain if and only if (0) is a prime ideal. (Note that the zero ring has no prime ideals, because the ideal (0) is the whole ring.) I is prime if and only if its set-theoretic complement is multiplicatively closed. {S} = {1}, we have Krull's theorem, and this recovers the maximal ideals of R. Another prototypical m-system is the set, {x, x2, x3, x4, ...}, of all positive powers of a non-nilpotent element. and (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) ∈ P + Q but its two factors are not. Alternatively, the quotient ring has zero divisors so it is not an integral domain and thus P + Q cannot be prime. (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.)

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 Richard Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.

Prime ideals for noncommutative rings

The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition "ideal-wise". Wolfgang Krull advanced this idea in 1928. The following content can be found in texts such as Goodearl's and Lam's. If R is a (possibly noncommutative) ring and P is a proper ideal of R, we say that P is prime if for any two ideals A and B of R: AB is contained in P, then at least one of A and B is contained in P. It can be shown that this definition is equivalent to the commutative one in commutative rings. It is readily verified that if an ideal of a noncommutative ring R satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal P satisfying the commutative definition of prime is sometimes called a completely prime ideal to distinguish it from other merely prime ideals in the ring. Completely prime ideals are prime ideals, but the converse is not true. For example, the zero ideal in the ring of n × n matrices over a field is a prime ideal, but it is not completely prime. 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. Equivalent formulations of the ideal P ≠ R being prime include the following properties: (a)(b) ⊆ P implies a ∈ P or b ∈ P . AB ⊆ P implies A ⊆ P or B ⊆ P . AB ⊆ P implies A ⊆ P or B ⊆ P . aRb ⊆ P , then a ∈ P or b ∈ P . Prime ideals in commutative rings are characterized by having multiplicatively closed complements in R, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset S ⊆ R is called an m-system if for any a and b in S, there exists r in R such that arb is in S. The following item can then be added to the list of equivalent conditions above: R∖P is an m-system.

Examples

Important facts

I1, ..., In is a collection of ideals of R with at most two members not prime, then if A is not contained in any Ij , it is also not contained in the union of I1, ..., In . In particular, A could be an ideal of R. {S} = {1}, we have Krull's theorem, and this recovers the maximal ideals of R. Another prototypical m-system is the set, {x, x2, x3, x4, ...}, of all positive powers of a non-nilpotent element. R∖P has another property beyond being an m-system. If xy is in R∖P , then both x and y must be in R∖P , since P is an ideal. A set that contains the divisors of its elements is called saturated. R∖S is a union of prime ideals of R.

Connection to maximality

Prime ideals can frequently be produced as maximal elements of certain collections of ideals. For example: