Prime ideal

File:A portion of the lattice of ideals of Z illustrating prime, semiprime and primary ideals SVG.svg

A Hasse diagram of a portion of the lattice of ideals of the integers <math>\Z.</math> The purple nodes indicate prime ideals. The purple and green nodes are semiprime ideals, and the purple and blue nodes are primary ideals.

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.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> 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 Template:Mvar of a commutative ring Template:Mvar is prime if it has the following two properties:

• If Template:Mvar and Template:Mvar are two elements of Template:Mvar such that their product Template:Math is an element of Template:Mvar, then Template:Math is in Template:Mvar or Template:Math is in Template:Mvar,
• Template:Mvar is not the whole ring Template:Mvar.

This generalizes the following property of prime numbers, known as Euclid's lemma: if Template:Math is a prime number and if Template:Math divides a product Template:Math of two integers, then Template:Math divides Template:Math or Template:Math divides Template:Math. We can therefore say

A positive integer Template:Mvar is a prime number if and only if <math>n\Z</math> is a prime ideal in <math>\Z.</math>

Examples

• A simple example: In the ring <math>R=\Z,</math> the subset of even numbers is a prime ideal.
• Given an integral domain <math>R</math>, any prime element <math>p \in R</math> generates a principal prime ideal <math>(p)</math>. For example, take an irreducible polynomial <math>f(x_1, \ldots, x_n)</math> in a polynomial ring <math>\mathbb{F}[x_1,\ldots,x_n]</math> over some field <math>\mathbb{F}</math>. Eisenstein's criterion for integral domains (hence UFDs) can be effective for determining if an element in a polynomial ring is irreducible.
• If Template:Mvar denotes the ring <math>\Complex[X,Y]</math> of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Template:Math is a prime ideal (see elliptic curve).
• In the ring <math>\Z[X]</math> of all polynomials with integer coefficients, the ideal generated by Template:Math and Template:Mvar is a prime ideal. The ideal consists of all polynomials constructed by taking Template:Math times an element of <math>\Z[X]</math> and adding it to Template:Mvar times another polynomial in <math>\Z[X]</math> (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.
• In any ring Template:Mvar, a maximal ideal is an ideal Template:Mvar that is maximal in the set of all proper ideals of Template:Mvar, i.e. Template:Mvar is contained in exactly two ideals of Template:Mvar, namely Template:Mvar itself and the whole ring Template:Mvar. 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. For the UFD Template:Nowrap Hilbert's Nullstellensatz states that every maximal ideal is of the form <math>(x_1-\alpha_1, \ldots, x_n-\alpha_n).</math>
• If Template:Mvar is a smooth manifold, Template:Mvar is the ring of smooth real functions on Template:Mvar, and Template:Mvar is a point in Template:Mvar, then the set of all smooth functions Template:Mvar with Template:Math forms a prime ideal (even a maximal ideal) in Template:Mvar.

Non-examples

• Consider the composition of the following two quotients

<math>\Complex[x,y] \to \frac{\Complex[x,y]}{(x^2 + y^2 - 1)} \to \frac{\Complex[x,y]}{(x^2 + y^2 - 1, x)}</math>

Although the first two rings are integral domains (in fact the first is a UFD) the last is not an integral domain since it is isomorphic to

<math>\frac{\Complex[x,y]}{(x^2 + y^2 - 1, x)} \cong \frac{\Complex[y]}{(y^2 - 1)} \cong \Complex\times\Complex</math>

since <math>(y^2 - 1)</math> factors into <math>(y - 1)(y + 1)</math>, which implies the existence of zero divisors in the quotient ring, preventing it from being isomorphic to <math>\Complex</math> and instead to non-integral domain <math>\Complex\times\Complex</math> (by the Chinese remainder theorem).

This shows that the ideal <math>(x^2 + y^2 - 1, x) \subset \Complex[x,y]</math> is not prime. (See the first property listed below.)

• Another non-example is the ideal <math>(2,x^2 + 5) \subset \Z[x]</math> since we have

<math>x^2+5 -2\cdot 3=(x-1)(x+1)\in (2,x^2+5)</math>

but neither <math>x-1</math> nor <math>x+1</math> are elements of the ideal.

Properties

• An ideal Template:Math in the ring Template:Mvar (with unity) is prime if and only if the factor ring Template:Math is an integral domain. In particular, a commutative ring (with unity) is an integral domain if and only if Template:Math is a prime ideal. (The zero ring has no prime ideals, because the ideal (0) is the whole ring.)
• An ideal Template:Math is prime if and only if its set-theoretic complement is multiplicatively closed.<ref>Template:Cite book</ref>
• 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.
• More generally, if Template:Mvar is any multiplicatively closed set in Template:Mvar, then a lemma essentially due to Krull shows that there exists an ideal of Template:Mvar maximal with respect to being disjoint from Template:Mvar, and moreover the ideal must be prime. This can be further generalized to noncommutative rings (see below).<ref name="Lam">Lam First Course in Noncommutative Rings, p. 156</ref> In the case Template:Math we have Krull's theorem, and this recovers the maximal ideals of Template:Mvar. Another prototypical m-system is the set, Template:Math of all positive powers of a non-nilpotent element.
• The preimage of a prime ideal under a ring homomorphism is a prime ideal. The analogous fact is not always true for maximal ideals, which is one reason algebraic geometers define the spectrum of a ring to be its set of prime rather than maximal ideals; one wants a homomorphism of rings to give a map between their spectra.
• The set of all prime ideals (called the spectrum of a ring) contains minimal elements (called minimal prime ideals). Geometrically, these correspond to irreducible components of the spectrum.
• The sum of two prime ideals is not necessarily prime. For an example, consider the ring <math>\Complex[x,y]</math> with prime ideals Template:Math and Template:Math (the ideals generated by Template:Math and Template:Math respectively). Their sum Template:Math however is not prime: Template:Math but its two factors are not. Alternatively, the quotient ring has zero divisors so it is not an integral domain and thus Template:Math cannot be prime.
• Not every ideal which cannot be factored into two ideals is a prime ideal; e.g. <math> (x,y^2)\subset \mathbb{R}[x,y]</math> cannot be factored but is not prime.
• In a commutative ring Template:Mvar with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal Template:Math is prime, then the ring Template:Mvar is an integral domain. If Template:Mvar is any non-zero element of Template:Mvar and the ideal Template:Math is prime, then it contains Template:Mvar and then Template:Mvar 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.

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.<ref>Krull, Wolfgang, Primidealketten in allgemeinen Ringbereichen, Sitzungsberichte Heidelberg. Akad. Wissenschaft (1928), 7. Abhandl.,3-14.</ref> The following content can be found in texts such as Goodearl's<ref>Goodearl, An Introduction to Noncommutative Noetherian Rings</ref> and Lam's.<ref>Lam, First Course in Noncommutative Rings</ref> If Template:Mvar is a (possibly noncommutative) ring and Template:Mvar is a proper ideal of Template:Mvar, we say that Template:Mvar is prime if for any two ideals Template:Mvar and Template:Mvar of Template:Mvar:

• If the product of ideals Template:Math is contained in Template:Mvar, then at least one of Template:Mvar and Template:Mvar is contained in Template:Mvar.

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 Template:Mvar satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal Template:Mvar 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 Template:Math 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 <math>\Z</math> "Template:Mvar is contained in Template:Mvar" is another way of saying "Template:Mvar divides Template:Mvar", and the unit ideal Template:Mvar represents unity.

Equivalent formulations of the ideal Template:Math being prime include the following properties:

• For all Template:Mvar and Template:Mvar in Template:Mvar, Template:Math implies Template:Math or Template:Math.
• For any two right ideals of Template:Mvar, Template:Math implies Template:Math or Template:Math.
• For any two left ideals of Template:Mvar, Template:Math implies Template:Math or Template:Math.
• For any elements Template:Mvar and Template:Mvar of Template:Mvar, if Template:Math, then Template:Math or Template:Math.

Prime ideals in commutative rings are characterized by having multiplicatively closed complements in Template:Mvar, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset Template:Math is called an m-system if for any Template:Mvar and Template:Mvar in Template:Mvar, there exists Template:Mvar in Template:Mvar such that Template:Math is in Template:Mvar.<ref>Obviously, multiplicatively closed sets are m-systems.</ref> The following item can then be added to the list of equivalent conditions above:

• The complement Template:Math is an m-system.

Examples

• Any primitive ideal is prime.
• 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 Template:Mvar is a nonzero Template:Mvar-module, and Template:Mvar is a maximal element in the poset of annihilator ideals of submodules of Template:Mvar, then Template:Mvar is prime.

Important facts

• Prime avoidance lemma. If Template:Mvar is a commutative ring, and Template:Mvar is a subring (possibly without unity), and Template:Math is a collection of ideals of Template:Mvar with at most two members not prime, then if Template:Mvar is not contained in any Template:Math, it is also not contained in the union of Template:Math.<ref>Jacobson Basic Algebra II, p. 390</ref> In particular, Template:Mvar could be an ideal of Template:Mvar.
• If Template:Mvar is any m-system in Template:Mvar, then a lemma essentially due to Krull shows that there exists an ideal Template:Mvar of Template:Mvar maximal with respect to being disjoint from Template:Mvar, and moreover the ideal Template:Mvar must be prime (the primality Template:Mvar can be proved as follows: if <math>a, b\not\in I</math>, then there exist elements <math>s, t\in S</math> such that <math>s\in I+(a), t\in I+(b)</math> by the maximal property of Template:Mvar. Now, if <math>(a)(b)\subset I</math>, then <math>st\in (I+(a))(I+(b))\subset I+(a)(b)\subset I</math>, which is a contradiction).<ref name="Lam"/> In the case Template:Math we have Krull's theorem, and this recovers the maximal ideals of Template:Mvar. Another prototypical m-system is the set, Template:Math of all positive powers of a non-nilpotent element.
• For a prime ideal Template:Mvar, the complement Template:Math has another property beyond being an m-system. If xy is in Template:Math, then both Template:Mvar and Template:Mvar must be in Template:Math, since Template:Mvar is an ideal. A set that contains the divisors of its elements is called saturated.
• For a commutative ring Template:Mvar, there is a kind of converse for the previous statement: If Template:Mvar is any nonempty saturated and multiplicatively closed subset of Template:Mvar, the complement Template:Math is a union of prime ideals of Template:Mvar.<ref>Kaplansky Commutative rings, p. 2</ref>
• The intersection of members of a descending chain of prime ideals is a prime ideal, and in a commutative ring the union of members of an ascending chain of prime ideals is a prime ideal. With Zorn's Lemma, these observations imply that the poset of prime ideals of a commutative ring (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 Template:Mvar-module Template:Mvar is prime.
• In a commutative ring, an ideal maximal with respect to being non-principal is prime.<ref>Kaplansky Commutative rings, p. 10, Ex 10.</ref>
• In a commutative ring, an ideal maximal with respect to being not countably generated is prime.<ref>Kaplansky Commutative rings, p. 10, Ex 11.</ref>

See also

• Radical ideal
• Maximal ideal
• Dedekind–Kummer theorem
• Residue field

Further reading

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