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In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, a prime ideal is a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
of a ring that shares many important properties of a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
in the ring of
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the
zero ideal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Primitive ideals are prime, and prime ideals are both
primary Primary or primaries may refer to: Arts, entertainment, and media Music Groups and labels * Primary (band), from Australia * Primary (musician), hip hop musician and record producer from South Korea * Primary Music, Israeli record label Works * ...
and
semiprime In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
.


Prime ideals for commutative rings

An
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
of a
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
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 In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A na ...
: if is a prime number and if
divides In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
a product of two
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, then divides or divides . We can therefore say :A positive integer is a prime number
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
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. * Given an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
R, any
prime element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
p \in R generates a
principal Principal may refer to: Title or rank * Principal (academia) The principal is the chief executive and the chief academic officer of a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher ...
prime ideal (p).
Eisenstein's criterionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
for integral domains (hence UFDs) is an effective tool for determining whether or not an element in a
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
is irreducible. For example, take an irreducible polynomial f(x_1, \ldots, x_n) in a polynomial ring \mathbb _1,\ldots,x_n/math> over some
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
\mathbb. * If denotes the ring \Complex ,Y/math> of
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s in two variables with
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, then the ideal generated by the polynomial is a prime ideal (see
elliptic curve In , an elliptic curve is a , , of one, on which there is a specified point ''O''. An elliptic curve is defined over a ''K'' and describes points in ''K''2, the of ''K'' with itself. If the field's is different from 2 and 3, then the curv ...
). * In the ring \Z /math> of all polynomials with integer coefficients, the ideal generated by and is a prime ideal. It consists of all those polynomials whose constant coefficient is even. * In any ring , a
maximal idealIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
is an ideal that is maximal in the set of all
proper ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloqu ...
s of , i.e. is
contained in
contained in
exactly two ideals of , namely itself and the whole ring . Every maximal ideal is in fact prime. In a
principal ideal domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
every nonzero prime ideal is maximal, but this is not true in general. For the UFD
Hilbert's Nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''Satz ' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

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 In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
, is the ring of smooth
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
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 Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
of the following two quotients ::\Complex ,y\to \frac \to \frac :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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
to ::\frac \cong \frac \cong \Complex\times\Complex :showing that the ideal (x^2 + y^2 - 1, x) \subset \Complex ,y/math> is not prime. (See the first property listed below.) * Another non-example is the ideal (2,x^2 + 5) \subset \Z /math> 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 Unity may refer to: Buildings * Unity Building The Unity Building, in Oregon, Illinois, is a historic building in that city's Oregon Commercial Historic District. As part of the district the Oregon Unity Building has been listed on the National R ...
) is prime if and only if the
factor ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra. ...
is an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. In particular, a commutative ring (with unity) is an integral domain if and only if is a prime ideal. * An ideal is prime if and only if its set-theoretic
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
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 theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. * 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
disjoint
from , and moreover the ideal must be prime. This can be further generalized to noncommutative rings (see below).Lam ''First Course in Noncommutative Rings'', p. 156 In the case we have
Krull's theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, and this recovers the maximal ideals of . Another prototypical m-system is the set, of all positive powers of a non-
nilpotent In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
element. * The
preimage In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of a prime ideal under a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
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 In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
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 In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
) 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 ,y/math> 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 divisor In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s 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 ,y/math> 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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. In a UFD, every nonzero prime ideal contains a prime element.


Uses

One use of prime ideals occurs in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models expla ...
, into a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
and can thus define generalizations of varieties called schemes, which find applications not only in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, but also in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
. The introduction of prime ideals in
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
was a major step forward: it was realized that the important property of unique factorisation expressed in the
fundamental theorem of arithmetic In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
does not hold in every ring of
algebraic integer In algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...
s, but a substitute was found when
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
replaced elements by ideals and prime elements by prime ideals; see
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...
.


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 Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...
advanced this idea in 1928. The following content can be found in texts such as Goodearl's and Lam's. 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 Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a categorical or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...
is not true. For example, the zero ideal in the ring of
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
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 numberIn number theory an ideal number is an algebraic integer which represents an ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated ...
s, 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 In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by includ ...

nonempty
subset is called an m-system if for any and in , there exists in such that is in . 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 ringIn abstract algebra, a nonzero ring ''R'' is a prime ring if for any two elements ''a'' and ''b'' of ''R'', ''arb = 0'' for all ''r'' in ''R'' implies that either ''a = 0'' or ''b = 0''. This definition can be regarded as a simultaneous generaliza ...
if and only if the zero ideal is a prime ideal, and moreover a ring is a
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
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 Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
, and is a maximal element in the
poset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of annihilator ideals of submodules of , then is prime.


Important facts

* Prime avoidance lemma. If is a commutative ring, and is a
subring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(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 . 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 . We can take r\in R with srt\in S. Now, if (a)(b)\subset I, then srt\in (I+(a))r(I+(b))\subset I+(a)(b)\subset I, which is a contradiction). In the case we have
Krull's theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, and this recovers the maximal ideals of . Another prototypical m-system is the set, of all positive powers of a non-
nilpotent In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 . *The
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
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 Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...
, 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. * In a commutative ring, an ideal maximal with respect to being not countably generated is prime.Kaplansky ''Commutative rings'', p. 10, Ex 11.


References


Further reading

* * * * * * {{DEFAULTSORT:Prime Ideal