Prime ideal explained

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.^{[1] [2]} 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 of a commutative ring is prime if it has the following two properties:

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

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

n\Z is a prime ideal in \Z.

Examples

• A simple example: In the ring

R = \Z, the subset of even numbers is a prime ideal.R , any prime element p \in R generates a principal prime ideal (p) . Eisenstein's criterion for integral domains (hence UFDs) is an effective tool for determining whether or not an element in a polynomial ring is irreducible. For example, take an irreducible polynomial f(x _{1,} \ldots, x _{n)} in a polynomial ring F [x _{1,\ldots,x} _{n]} over some field F .

• If denotes the ring

\Complex[X,Y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial is a prime ideal (see elliptic curve).

• In the ring

\Z[X] of all polynomials with integer coefficients, the ideal generated by and is a prime ideal. The ideal consists of all polynomials constructed by taking times an element of \Z[X] and adding it to 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.

• In any ring, a maximal ideal is an ideal that is maximal in the set of all proper ideals of, i.e. is contained in exactly two ideals of, namely itself and the whole ring . 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 Hilbert's Nullstellensatz states that every maximal ideal is of the form

(x _{1-\alpha} _{1,} \ldots, x _{n-\alpha} _{n).}

• If is a smooth manifold, is the ring of smooth real functions on, and is a point in, then the set of all smooth functions with forms a prime ideal (even a maximal ideal) in .

Non-examples

• Consider the composition of the following two quotients

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

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

(\Complex[x,y])/((x ^{2} + y ^{2} -1 , x)) \cong (\Complex[y])/((y ^{2} -1 )) \cong \Complex x \Complex

since

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

This shows that the ideal

(x ^{2} + y ^{2} - 1 , x) \subset \Complex[x,y] is not prime. (See the first property listed below.)

• Another non-example is the ideal

( 2 ,x ^{2} + 5 ) \subset \Z[x] since we havex ^{2+5} -2 ⋅ 3= (x -1 )(x +1 )\in ( 2 ,x ^{2+5)}

but neither

x -1 nor x +1 are elements of the ideal.

Properties

• An ideal in the ring (with unity) is prime if and only if the factor ring is an integral domain. In particular, a commutative ring (with unity) is an integral domain if and only if is a prime ideal. (Note that the zero ring has no prime ideals, because the ideal (0) is the whole ring.)
• An ideal is prime if and only if its set-theoretic complement is multiplicatively closed.^[3]
• 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 is any multiplicatively closed set in, then a lemma essentially due to Krull shows that there exists an ideal of maximal with respect to being disjoint from, and moreover the ideal must be prime. This can be further generalized to noncommutative rings (see below).^[4] In the case we have Krull's theorem, and this recovers the maximal ideals of . Another prototypical m-system is the set, 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

\Complex[x,y] with prime ideals and (the ideals generated by and respectively). Their sum however is not prime: but its two factors are not. Alternatively, the quotient ring has zero divisors so it is not an integral domain and thus cannot be prime.

• Not every ideal which cannot be factored into two ideals is a prime ideal; e.g.

(x,y ^{2)\subset} R [x,y] cannot be factored but is not prime.

• In a commutative ring with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal is prime, then the ring is an integral domain. If is any non-zero element of and the ideal is prime, then it contains and then 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.^[5] The following content can be found in texts such as Goodearl's^[6] and Lam's.^[7] If is a (possibly noncommutative) ring and is a proper ideal of, we say that is prime if for any two ideals and of :

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

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 satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal 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 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 " is contained in " is another way of saying " divides ", and the unit ideal represents unity.

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

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

Prime ideals in commutative rings are characterized by having multiplicatively closed complements in, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset is called an m-system if for any and in, there exists in such that is in .^[8] The following item can then be added to the list of equivalent conditions above:

• The complement 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 is a nonzero -module, and is a maximal element in the poset of annihilator ideals of submodules of, then is prime.

Important facts

• Prime avoidance lemma. If is a commutative ring, and is a subring (possibly without unity), and is a collection of ideals of with at most two members not prime, then if is not contained in any, it is also not contained in the union of .^[9] In particular, could be an ideal of .
• If is any m-system in, then a lemma essentially due to Krull shows that there exists an ideal of maximal with respect to being disjoint from, and moreover the ideal must be prime (the primality can be proved as follows: if

a, b\not\in I , then there exist elements s, t\in S such that s\in I + (a), t\in I + (b) by the maximal property of . Now, if (a)(b)\subset I , then st\in (I + (a))(I + (b))\subset I + (a)(b)\subset I , which is a contradiction).^[4] In the case we have Krull's theorem, and this recovers the maximal ideals of . Another prototypical m-system is the set, of all positive powers of a non-nilpotent element.

• For a prime ideal, the complement has another property beyond being an m-system. If xy is in, then both and must be in, since is an ideal. A set that contains the divisors of its elements is called saturated.
• For a commutative ring, there is a kind of converse for the previous statement: If is any nonempty saturated and multiplicatively closed subset of, the complement is a union of prime ideals of .^[10]
• 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 -module is prime.
• In a commutative ring, an ideal maximal with respect to being non-principal is prime.^[11]
• In a commutative ring, an ideal maximal with respect to being not countably generated is prime.^[12]

See also

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

Notes and References

1. Book: Dummit . David S. . Foote . Richard M. . Abstract Algebra . . 2004 . 3rd . 0-471-43334-9.
2. Book: Lang, Serge . Serge Lang . Algebra . . . 2002 . 0-387-95385-X.
3. Book: Reid, Miles . Miles Reid . Undergraduate Commutative Algebra . . 1996 . 0-521-45889-7.
4. Lam First Course in Noncommutative Rings, p. 156
5. Krull, Wolfgang, Primidealketten in allgemeinen Ringbereichen, Sitzungsberichte Heidelberg. Akad. Wissenschaft (1928), 7. Abhandl.,3-14.
6. Goodearl, An Introduction to Noncommutative Noetherian Rings
7. Lam, First Course in Noncommutative Rings
8. Obviously, multiplicatively closed sets are m-systems.
9. Jacobson Basic Algebra II, p. 390
10. Kaplansky Commutative rings, p. 2
11. Kaplansky Commutative rings, p. 10, Ex 10.
12. Kaplansky Commutative rings, p. 10, Ex 11.