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 :A positive integer is a prime number
if and only if n\Z is a prime ideal in \Z. The set of prime ideals of a commutative ring
R is known as its
(prime) spectrum and is denoted \mathrm{Spec}\ R. Depending on context, this terminology and notation are also used to refer to the set of prime ideals equipped with additional structures, a topology and a sheaf of rings, that make it a geometric object known as an affine
scheme.
Alternative Definition An equivalent and potentially easier to understand definition is as follows. Let be a
commutative ring. A proper
ideal of is
prime if it has the following property: • If and , then . This property is mathematically equivalent to the standard definition used above as it was derived using the contrapositive.
Examples • A simple example: In the ring R=\Z, the subset of
even numbers 2\mathbb{Z} (also denoted (2), meaning the ideal generated by 2) is a prime ideal. • Given an
integral domain R, any
prime element p \in R generates a
principal prime ideal (p). For example, take an irreducible polynomial f(x_1, \ldots, x_n) in a polynomial ring \mathbb{F}[x_1,\ldots,x_n] over some
field \mathbb{F}.
Eisenstein's criterion for integral domains (hence
UFDs) can be effective for determining if an element in a
polynomial ring is
irreducible. • 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 (2,X) 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 \frac{\Complex[x,y]}{(x^2 + y^2 - 1)} \to \frac{\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 ::\frac{\Complex[x,y]}{(x^2 + y^2 - 1, x)} \cong \frac{\Complex[y]}{(y^2 - 1)} \cong \Complex\times\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\times\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 have ::x^2+5 -2\cdot 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. (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. • 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). In the case {{math|
S {1},}} we have
Krull's theorem, and this recovers the maximal ideals of . Another prototypical m-system is the set, {{math|{
x,
x2,
x3,
x4, ...},}} 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 \mathbb{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==