# Prime ideal

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In algebra, a prime ideal is a subset of a ring which 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

An ideal P of a commutative ring R is prime if it has the following two properties:

• If a and b are two elements of R such that their product ab is an element of 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 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.

### 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 Template:Harv and Template:Harv. If R is a (possibly noncommutative) ring and P is an ideal in R other than R itself, we say that P is prime if for any two ideals A and B of R:

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 PR being prime include the following properties:

• For all a and b in R, (a)(b)⊆P implies aP or bP.
• For any two right ideals of R, ABP implies AP or BP.
• For any two left ideals of R, ABP implies AP or BP.
• For any elements a and b of R, if aRbP, then aP or bP.

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 SR 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:

• The complement R\P is an m-system.

### Examples

• As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals.
• A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal.
• Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R module, and P is a maximal element in the poset of annihilator ideals of submodules of A, then P is prime.

## Important facts

• Prime avoidance lemma: If R is a commutative ring, and A is a subring (possibly without unity), and 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.
• If S is any m-system in R, then a lemma essentially due to Krull shows that there exists an ideal of R maximal with respect to being disjoint from S, and moreover the ideal must be prime. In the case {S}={1}, we have Krull's theorem, and this recovers the maximal ideals of R. Another prototypical m-system is the set $\{x,x^{2},x^{3},x^{4},\dots \}$ of all positive powers of a non-nilpotent element.
• For a prime ideal P, the complement 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 which contains the divisors of its elements is called saturated.
• For a commutative ring R, there is a kind of converse for the previous statement: If S is any nonempty saturated and multiplicatively closed subset of R, the complement R\S is a union of prime ideals of R.
• The union of members of an ascending chain of prime ideals is a prime ideal, and the intersection of members of a descending chain of prime ideals is a prime ideal. With Zorn's Lemma, these observations imply that the poset of prime ideals (partially ordered by inclusion) has maximal and minimal elements.

## Connection to maximality

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

• An ideal maximal with respect to having empty intersection with a fixed m-system is prime.
• An ideal maximal among annihilators of submodules of a fixed R module M is prime.
• In a commutative ring, an ideal maximal with respect to being non-principal is prime.
• In a commutative ring, an ideal maximal with respect to being not countably generated is prime.