Editing Prime ideal (section)

==Prime ideals for commutative rings==

===Definition===
An [[ideal (ring theory)|ideal]] {{mvar|P}} of a [[commutative ring]] {{mvar|R}} is '''prime''' if it has the following two properties:

* If {{mvar|a}} and {{mvar|b}} are two elements of {{mvar|R}} such that their product {{math|''ab''}} is an element of {{mvar|P}}, then {{math|''a''}} is in {{mvar|P}} or {{math|''b''}} is in {{mvar|P}},
* {{mvar|P}} is not the whole ring {{mvar|R}}.

This generalizes the following property of prime numbers, known as [[Euclid's lemma]]: if {{math|''p''}} is a prime number and if {{math|''p''}} [[divides]] a product {{math|''ab''}} of two [[integer]]s, then {{math|''p''}} divides {{math|''a''}} or {{math|''p''}} divides {{math|''b''}}. We can therefore say

:A positive integer {{mvar|n}} 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 [[parity (mathematics)|even]] numbers is a prime ideal.
* Given an [[integral domain]] <math>R</math>, any [[prime element]] <math>p \in R</math> generates a [[principal ideal domain|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 (mathematics)|field]] <math>\mathbb{F}</math>. [[Eisenstein's criterion]] for integral domains (hence [[Unique factorization domain|UFDs]]) can be effective for determining if an element in a [[polynomial ring]] is [[irreducible polynomial|irreducible]].
* If {{mvar|R}} denotes the ring <math>\Complex[X,Y]</math> of [[polynomial]]s in two variables with [[complex number|complex]] [[coefficient]]s, then the ideal generated by the polynomial {{math|''Y''<sup>&thinsp;2</sup> − ''X''<sup>&thinsp;3</sup> − ''X'' − 1}} is a prime ideal (see [[elliptic curve]]).
* In the ring <math>\Z[X]</math> of all polynomials with integer coefficients, the ideal generated by {{math|2}} and {{mvar|X}} is a prime ideal. The ideal consists of all polynomials constructed by taking {{math|2}} times an element of <math>\Z[X]</math> and adding it to {{mvar|X}} 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 {{mvar|R}}, a '''[[maximal ideal]]''' is an ideal {{mvar|M}} that is [[maximal element|maximal]] in the set of all [[proper ideal]]s of {{mvar|R}}, i.e. {{mvar|M}} is [[subset|contained in]] exactly two ideals of {{mvar|R}}, namely {{mvar|M}} itself and the whole ring {{mvar|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. For the UFD {{nowrap|<math>\Complex[x_1,\ldots,x_n]</math>,}} [[Hilbert's Nullstellensatz]] states that every maximal ideal is of the form <math>(x_1-\alpha_1, \ldots, x_n-\alpha_n).</math>
* If {{mvar|M}} is a [[Manifold#Differentiable manifolds|smooth manifold]], {{mvar|R}} is the ring of smooth [[real number|real]] functions on {{mvar|M}}, and {{mvar|x}} is a point in {{mvar|M}}, then the set of all smooth functions {{mvar|f}} with {{math|''f''&thinsp;(''x'') {{=}} 0}} forms a prime ideal (even a maximal ideal) in {{mvar|R}}.

=== Non-examples ===
* Consider the [[function composition|composition]] of the following two [[quotient ring|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 [[ring homomorphism|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_divisor|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#Statement|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 {{math|''I''}} in the ring {{mvar|R}} (with [[unital ring|unity]]) is prime if and only if the [[factor ring]] {{math|''R''/''I''}} is an [[integral domain]]. In particular, a commutative ring (with unity) is an integral domain if and only if {{math|(0)}} is a prime ideal.  (The [[zero ring]] has no prime ideals, because the ideal (0) is the whole ring.)
* An ideal {{math|''I''}} is prime if and only if its set-theoretic [[complement (set theory)|complement]] is [[multiplicatively closed set|multiplicatively closed]].<ref>{{cite book | last=Reid | first=Miles | author-link=Miles Reid | title=Undergraduate Commutative Algebra | publisher=[[Cambridge University Press]] | year=1996 | isbn=0-521-45889-7}}</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 {{mvar|S}} is any multiplicatively closed set in {{mvar|R}}, then a lemma essentially due to Krull shows that there exists an ideal of {{mvar|R}} maximal with respect to being [[disjoint sets|disjoint]] from {{mvar|S}}, 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 {{math|''S'' {{=}} {1},}} we have [[Krull's theorem]], and this recovers the maximal ideals of {{mvar|R}}. Another prototypical m-system is the set, {{math|{''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ''x''<sup>4</sup>, ...},}} 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 ideal]]s, 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 ideal]]s). 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 {{math|''P'' {{=}} (''x''<sup>2</sup> + ''y''<sup>2</sup> − 1)}} and {{math|''Q'' {{=}} (''x'')}} (the ideals generated by {{math|''x''<sup>2</sup> + ''y''<sup>2</sup> − 1}} and {{math|''x''}} respectively). Their sum {{math|''P'' + ''Q'' {{=}} (''x''<sup>2</sup> + ''y''<sup>2</sup> − 1, ''x'') {{=}} (''y''<sup>2</sup> − 1, ''x'')}} however is not prime: {{math|''y''<sup>2</sup> − 1 {{=}} (''y'' − 1)(''y'' + 1) ∈ ''P'' + ''Q''}} but its two factors are not. Alternatively, the quotient ring has [[zero divisor]]s so it is not an integral domain and thus {{math|''P'' + ''Q''}} 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 {{mvar|R}} with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal {{math|(0)}} is prime, then the ring {{mvar|R}} is an integral domain. If {{mvar|q}} is any non-zero element of {{mvar|R}} and the ideal {{math|(''q''<sup>2</sup>)}} is prime, then it contains {{mvar|q}} and then {{mvar|q}} is [[unit (ring theory)|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 of a ring|spectrum]], into a [[topological space]] and can thus define generalizations of varieties called [[scheme (mathematics)|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 integer]]s, but a substitute was found when [[Richard Dedekind]] replaced elements by ideals and prime elements by prime ideals; see [[Dedekind domain]].