prime ideal

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In algebra, a prime ideal is a 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. 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, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive ident ...
.
Primitive ideal In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals ...
s are prime, and prime ideals are both primary and semiprime.

# Prime ideals for commutative rings

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 with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
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 In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as ...
: 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 $\Z.$

## Examples

* A simple example: In the ring $R=\Z,$ the subset of even numbers is a prime ideal. * Given an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
$R$, any prime element $p \in R$ generates a principal prime ideal $\left(p\right)$. 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 $f\left(x_1, \ldots, x_n\right)$ in a polynomial ring
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
coefficients, then the ideal generated by the polynomial is a prime ideal (see elliptic curve). * In the ring
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
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 $\left(x_1-\alpha_1, \ldots, x_n-\alpha_n\right).$ * 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 :: :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 ::$\frac \cong \frac \cong \Complex\times\Complex$ :showing that the ideal

## Properties

* An ideal in the ring (with unity) is prime if and only if the factor ring is an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
. In particular, a commutative ring (with unity) is an integral domain if and only if is a prime ideal. (Note that the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which fo ...
has no prime ideals, because the ideal (0) is the whole ring.) * An ideal is prime if and only if its set-theoretic
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
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).Lam ''First Course in Noncommutative Rings'', p. 156 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 In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
element. * The
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
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 In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
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 commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
) 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

## 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 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. The word was first used scientifically in optics to describe the rainbow of color ...
, into a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
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 Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject. Krull was born and went to school in Baden-Baden. ...
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 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 number In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the rin ...
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 having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
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 In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals ...
is prime. * As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals. * A ring is a
prime ring In 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 generaliz ...
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 . 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 $s\in I+\left(a\right), t\in I+\left(b\right)$ by the maximal property of . We can take $r\in R$ with $srt\in S$. Now, if $\left(a\right)\left(b\right)\subset I$, then $srt\in \left(I+\left(a\right)\right)r\left(I+\left(b\right)\right)\subset I+\left(a\right)\left(b\right)\subset I$, which is a contradiction). 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 In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
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 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. * In a commutative ring, an ideal maximal with respect to being not countably generated is prime.Kaplansky ''Commutative rings'', p. 10, Ex 11.