Editing Prime ideal

{{Short description|Ideal in a ring which has properties similar to prime elements}}
{{about|ideals in [[ring theory]]|prime ideals in order theory|Ideal (order theory)#Prime ideals}}
{{distinguish|text=[[Primary ideal]]}}
[[File:A portion of the lattice of ideals of Z illustrating prime, semiprime and primary ideals SVG.svg|thumb|right|upright=1.3|A [[Hasse diagram]] of a portion of the [[lattice (order)|lattice]] of [[ideal (ring theory)|ideals]] of the integers <math>\Z.</math> The purple nodes indicate prime ideals. The purple and green nodes are [[semiprime ideal]]s, and the purple and blue nodes are [[primary ideal]]s.]] In [[algebra]], a '''prime ideal''' is a [[subset]] of a [[ring (mathematics)|ring]] that shares many important properties of a [[prime number]] in the ring of [[Integer#Algebraic properties|integers]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | author-link=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> The prime ideals for the integers are the sets that contain all the [[multiple (mathematics)|multiples]] of a given prime number, together with the [[zero ideal]].

[[Primitive ideal]]s are prime, and prime ideals are both [[primary ideal|primary]] and [[semiprime ideal|semiprime]].

==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]].

==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 {{mvar|R}} is a (possibly noncommutative) ring and {{mvar|P}} is a proper ideal of {{mvar|R}}, we say that {{mvar|P}} is '''prime''' if for any two ideals {{mvar|A}} and {{mvar|B}} of {{mvar|R}}:

* If the product of ideals {{math|''AB''}} is contained in {{mvar|P}}, then at least one of {{mvar|A}} and {{mvar|B}} is contained in {{mvar|P}}.

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 {{mvar|R}} satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal {{mvar|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 (logic)|converse]] is not true.  For example, the zero ideal in the ring of {{math|''n''&thinsp;×&thinsp;''n''}} [[matrix (mathematics)|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 number]]s, as for the ring <math>\Z</math> "{{mvar|A}} is contained in {{mvar|P}}" is another way of saying "{{mvar|P}} divides {{mvar|A}}", and the unit ideal {{mvar|R}} represents unity.

Equivalent formulations of the ideal {{math|''P'' ≠ ''R''}} being prime include the following properties:
* For all {{mvar|a}} and {{mvar|b}} in {{mvar|R}}, {{math|(''a'')(''b'') ⊆ ''P''}} implies {{math|''a'' ∈ ''P''}} or {{math|''b'' ∈ ''P''}}.
* For any two ''right'' ideals of {{mvar|R}}, {{math|''AB'' ⊆ ''P''}} implies {{math|''A'' ⊆ ''P''}} or {{math|''B'' ⊆ ''P''}}.
* For any two ''left'' ideals of {{mvar|R}}, {{math|''AB'' ⊆ ''P''}} implies {{math|''A'' ⊆ ''P''}} or {{math|''B'' ⊆ ''P''}}.
* For any elements {{mvar|a}} and {{mvar|b}} of {{mvar|R}}, if {{math|''aRb'' ⊆ ''P''}}, then {{math|''a'' ∈ ''P''}} or {{math|''b'' ∈ ''P''}}.

Prime ideals in commutative rings are characterized by having multiplicatively closed [[complement (set theory)|complements]] in {{mvar|R}}, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A [[empty set|nonempty]] subset {{math|''S'' ⊆ ''R''}} is called an '''m-system''' if for any {{mvar|a}} and {{mvar|b}} in {{mvar|S}}, there exists {{mvar|r}} in {{mvar|R}} such that {{math|''arb''}} is in {{mvar|S}}.<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 {{math|''R''∖''P''}} 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 [[integral domain|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 {{mvar|A}} is a nonzero {{mvar|R}}-[[module (mathematics)|module]], and {{mvar|P}} is a maximal element in the [[poset]] of [[Annihilator (ring theory)|annihilator]] ideals of submodules of {{mvar|A}}, then {{mvar|P}} is prime.

==Important facts==
*'''[[Prime avoidance lemma]].''' If {{mvar|R}} is a commutative ring, and {{mvar|A}} is a [[subring]] (possibly without unity), and {{math|''I''<sub>1</sub>, ..., ''I<sub>n</sub>''}} is a collection of ideals of {{mvar|R}} with at most two members not prime, then if {{mvar|A}} is not contained in any {{math|''I<sub>j</sub>''}}, it is also not contained in the [[union (set theory)|union]] of {{math|''I''<sub>1</sub>, ..., ''I<sub>n</sub>''}}.<ref>Jacobson ''Basic Algebra II'', p. 390</ref> In particular, {{mvar|A}} could be an ideal of {{mvar|R}}.
* If {{mvar|S}} is any m-system in {{mvar|R}}, then a lemma essentially due to Krull shows that there exists an ideal {{mvar|I}} of {{mvar|R}} maximal with respect to being disjoint from {{mvar|S}}, and moreover the ideal {{mvar|I}} must be prime (the primality {{mvar|I}} can be [[mathematical proof|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 {{mvar|I}}. 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 {{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.
* For a prime ideal {{mvar|P}}, the complement {{math|''R''∖''P''}} has another property beyond being an m-system. If ''xy'' is in {{math|''R''∖''P''}}, then both {{mvar|x}} and {{mvar|y}} must be in {{math|''R''∖''P''}}, since {{mvar|P}} is an ideal. A set that contains the divisors of its elements is called '''saturated'''.
* For a commutative ring {{mvar|R}}, there is a kind of converse for the previous statement: If {{mvar|S}} is any nonempty saturated and multiplicatively closed subset of {{mvar|R}}, the complement {{math|''R''∖''S''}} is a union of prime ideals of {{mvar|R}}.<ref>Kaplansky ''Commutative rings'', p. 2</ref>
*The [[intersection (set theory)|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 [[Annihilator (ring theory)|annihilators]] of submodules of a fixed {{mvar|R}}-module {{mvar|M}} 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]]

==References==
{{Reflist}}

==Further reading==
*{{citation|author1=Goodearl, K. R.|author2=Warfield, R. B. Jr.|title=An introduction to noncommutative Noetherian rings| series=London Mathematical Society Student Texts |volume=61 |edition=2 |publisher=Cambridge University Press | place=Cambridge  |year=2004  |pages=xxiv+344<!--|isbn=0-521-83687-5-->|isbn=0-521-54537-4 |mr=2080008 |doi=10.1017/CBO9780511841699}}
*{{citation|author=Jacobson, Nathan|title=Basic algebra. II|edition=2|publisher=W. H. Freeman and Company |place=New York|year=1989 |pages=xviii+686   |isbn=0-7167-1933-9 |mr=1009787}}
*{{citation|author=Kaplansky, Irving |title=Commutative rings |publisher=Allyn and Bacon Inc. |place=Boston, Mass.|year=1970   |pages=x+180   |mr=0254021 }}
*{{citation|author=Lam, T. Y.|author-link=Tsit Yuen Lam |title=A first course in noncommutative rings |series=Graduate Texts in Mathematics|volume=131 |edition=2nd |publisher=Springer-Verlag |place=New York |year=2001 |pages=xx+385 |isbn=0-387-95183-0   |mr=1838439 | zbl=0980.16001 |doi=10.1007/978-1-4419-8616-0}}
*{{citation|author1=Lam, T. Y. | author1-link=Tsit Yuen Lam  |author2=Reyes, Manuel L. |title=A prime ideal principle in commutative algebra |journal=J. Algebra |volume=319 |year=2008 |number=7 |pages=3006–3027 |issn=0021-8693 |mr=2397420 | zbl=1168.13002 |doi=10.1016/j.jalgebra.2007.07.016|doi-access=free }}
* {{springer|title=Prime ideal|id=p/p074510}}

{{DEFAULTSORT:Prime Ideal}}
[[Category:Prime ideals| ]]