TheInfoList

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 general consensus abo ...
, specifically
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), rings, field (mathema ...
, an integral domain is a nonzero
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 ...
in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s and provide a natural setting for studying
divisibility 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). ...
. In an integral domain, every nonzero element ''a'' has the
cancellation property 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 ...
, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a
multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
" for the general case including noncommutative rings. Some sources, notably
Lang Lang may refer to: *Lang (surname) Lang is a surname of Germanic origin, closely related to Lange, Laing and Long, all of which mean "tall". "Lang" (Láng) is also a surname in Hungary, a cognate of the Hungarian word for "flame." https://www ... , use the term entire ring for integral domain. Some specific kinds of integral domains are given with the following chain of class inclusions:

# Definition

An ''integral domain'' is a nonzero
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 ...
in which the product of any two nonzero elements is nonzero. Equivalently: * An integral domain is a nonzero commutative ring with no nonzero
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. * An integral domain is a commutative ring in which 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 ...
is a
prime ideal 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. ...
. * An integral domain is a nonzero commutative ring for which every non-zero element is cancellable under multiplication. * An integral domain is a ring for which the set of nonzero elements is a commutative
monoid 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 (mathematic ...
under multiplication (because a monoid must be closed under multiplication). * An integral domain is a nonzero commutative ring in which for every nonzero element ''r'', the function that maps each element ''x'' of the ring to the product ''xr'' is
injective 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 ...
. Elements ''r'' with this property are called ''regular'', so it is equivalent to require that every nonzero element of the ring be regular. * An integral domain is a ring that 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 a
subring In mathematics, a subring of ''R'' is a subset of a ring (mathematics), ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ' ...
of a
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 ...
. (Given an integral domain, one can embed it in its
field of fractions In abstract algebra, the field of fractions of an integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
.)

# Examples

* The archetypical example is the ring $\Z$ of all
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s. * Every
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 ...
is an integral domain. For example, the field $\R$ of all
real number 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). ...
s is an integral domain. Conversely, every Artinian integral domain is a field. In particular, all finite integral domains are
finite field 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 ...
s (more generally, by
Wedderburn's little 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 ...
, finite domains are
finite field 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 ...
s). The ring of integers $\Z$ provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as: ::$\Z \supset 2\Z \supset \cdots \supset 2^n\Z \supset 2^\Z \supset \cdots$ * Rings 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 ... s are integral domains if the coefficients come from an integral domain. For instance, the ring
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 ... coefficients. * The previous example can be further exploited by taking quotients from prime ideals. For example, the ring corresponding to a plane
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 ...
is an integral domain. Integrality can be checked by showing $y^2 - x\left(x-1\right)\left(x-2\right)$is an
irreducible polynomial 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 ge ...
. * The ring
sqrt /math> is an integral domain for any non-square integer $n$. If $n > 0$, then this ring is always a subring of $\R$, otherwise, it is a subring of $\Complex.$ * The ring of
p-adic integers group In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real number, real and complex number syst ...
$\Z_p$ is an integral domain. * If $U$ is a connected
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of the
complex plane Image:Complex conjugate picture.svg, Geometric representation of ''z'' and its conjugate ''z̅'' in the complex plane. The distance along the light blue line from the origin to the point ''z'' is the ''modulus'' or ''absolute value'' of ''z''. The ... $\Complex$, then the ring $\mathcal\left(U\right)$ consisting of all
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
s is an integral domain. The same is true for rings of
analytic function 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 ge ...
s on connected open subsets of analytic
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 surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ... s. * A
regular local ringIn commutative algebra Commutative algebra is the branch of 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 nu ...
is an integral domain. In fact, a regular local ring is a UFD.

# Non-examples

The following rings are ''not'' integral domains. * The
zero ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
(the ring in which $0=1$). * The
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
$\Z/m\Z$ when ''m'' is a
composite number A composite number is a positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
. Indeed, choose a proper factorization $m = xy$ (meaning that $x$ and $y$ are not equal to $1$ or $m$). Then $x \not\equiv 0 \bmod$ and $y \not\equiv 0 \bmod$, but $xy \equiv 0 \bmod$. * A product of two nonzero commutative rings. In such a product $R \times S$, one has $\left(1,0\right) \cdot \left(0,1\right) = \left(0,0\right)$. * The
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
for any $n \in \mathbb$. The images of $x+n$ and $x-n$ are nonzero, while their product is 0 in this ring. * The ring of ''n'' × ''n''
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 any nonzero ring when ''n'' ≥ 2. If $M$ and $N$ are matrices such that the image of $N$ is contained in the kernel of $M$, then $MN = 0$. For example, this happens for $M = N = \left(\begin 0 & 1 \\ 0 & 0 \end\right)$. * The quotient ring for any field $k$ and any non-constant polynomials
prime ideal 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. ...
. The geometric interpretation of this result is that the zeros of form an
affine algebraic set Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certai ...
that is not irreducible (that is, not an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
) in general. The only case where this algebraic set may be irreducible is when is a power of an
irreducible polynomial 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 ge ...
, which defines the same algebraic set. * The ring of
continuous function 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 gen ...
s on the
unit interval 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 ...
. Consider the functions :: :Neither $f$ nor $g$ is everywhere zero, but $fg$ is. * The
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space that can be thought of as the ''space of all tensors'' that can be built from vectors from its constituent spac ... $\Complex \otimes_ \Complex$. This ring has two non-trivial
idempotent Idempotence (, ) is the property of certain operations in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
s, $e_1 = \tfrac\left(1 \otimes 1\right) - \tfrac\left(i \otimes i\right)$ and $e_2 = \tfrac\left(1 \otimes 1\right) + \tfrac\left(i \otimes i\right)$. They are orthogonal, meaning that $e_1e_2 = 0$, and hence $\Complex \otimes_ \Complex$ is not a domain. In fact, there is an isomorphism $\Complex \times \Complex \to \Complex \otimes_ \Complex$ defined by $\left(z, w\right) \mapsto z \cdot e_1 + w \cdot e_2$. Its inverse is defined by $z \otimes w \mapsto \left(zw, z\overline\right)$. This example shows that a
fiber product In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
of irreducible affine schemes need not be irreducible.

# Divisibility, prime elements, and irreducible elements

In this section, ''R'' is an integral domain. Given elements ''a'' and ''b'' of ''R'', one says that ''a'' ''divides'' ''b'', or that ''a'' is a ''
divisor 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 a ...
'' of ''b'', or that ''b'' is a ''multiple'' of ''a'', if there exists an element ''x'' in ''R'' such that . The ''
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
s'' of ''R'' are the elements that divide 1; these are precisely the invertible elements in ''R''. Units divide all other elements. If ''a'' divides ''b'' and ''b'' divides ''a'', then ''a'' and ''b'' are associated elements or associates. Equivalently, ''a'' and ''b'' are associates if for some
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
''u''. An ''
irreducible elementIn abstract algebra, a non-zero non-unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatri ...
'' is a nonzero non-unit that cannot be written as a product of two non-units. A nonzero non-unit ''p'' is 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 ...
'' if, whenever ''p'' divides a product ''ab'', then ''p'' divides ''a'' or ''p'' divides ''b''. Equivalently, an element ''p'' is prime if and only if the
principal ideal 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 ...
(''p'') is a nonzero prime ideal. Both notions of irreducible elements and prime elements generalize the ordinary definition 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 natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s in the ring $\Z,$ if one considers as prime the negative primes. Every prime element is irreducible. The converse is not true in general: for example, in the
quadratic integer In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are ...
ring
GCD domain In mathematics, a GCD domain is an integral domain ''R'' with the property that any two elements have a greatest common divisor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number th ...
), an irreducible element is a prime element. While
unique factorization 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 t ...
does not hold in
ideals 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 considered ...
. See
Lasker–Noether theorem In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related ...
.

# Properties

* A commutative ring ''R'' is an integral domain if and only if the ideal (0) of ''R'' is a prime ideal. * If ''R'' is a commutative ring and ''P'' is 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 ...
in ''R'', then the
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
''R/P'' is an integral domain if and only if ''P'' is a
prime ideal 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. ...
. * Let ''R'' be an integral domain. Then the
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). ...
s over ''R'' (in any number of indeterminates) are integral domains. This is in particular the case if ''R'' is a
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 ...
. * The cancellation property holds in any integral domain: for any ''a'', ''b'', and ''c'' in an integral domain, if ''a'' ≠ ''0'' and ''ab'' = ''ac'' then ''b'' = ''c''. Another way to state this is that the function ''x'' ''ax'' is injective for any nonzero ''a'' in the domain. * The cancellation property holds for ideals in any integral domain: if ''xI'' = ''xJ'', then either ''x'' is zero or ''I'' = ''J''. * An integral domain is equal to the intersection of its localizations at maximal ideals. * An
inductive limitIn mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be Group (mathematics), groups, Ring (mathematics), rings, Vector space, ...
of integral domains is an integral domain. *If $A, B$ are integral domains over an algebraically closed field ''k'', then $A \otimes_k B$ is an integral domain. This is a consequence of
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 ... ,Proof: First assume ''A'' is finitely generated as a ''k''-algebra and pick a $k$-basis $g_i$ of $B$. Suppose $\sum f_i \otimes g_i \sum h_j \otimes g_j = 0$ (only finitely many $f_i, h_j$ are nonzero). For each maximal ideal $\mathfrak$ of $A$, consider the ring homomorphism $A \otimes_k B \to A/\mathfrak \otimes_k B = k \otimes_k B \simeq B$. Then the image is $\sum \overline g_i \sum \overline g_i = 0$ and thus either $\sum \overline g_i = 0$ or $\sum \overline g_i = 0$ and, by linear independence, $\overline = 0$ for all $i$ or $\overline = 0$ for all $i$. Since $\mathfrak$ is arbitrary, we have $(\sum f_iA) (\sum h_iA) \subset \operatorname(A) =$ the intersection of all maximal ideals $= \left(0\right)$ where the last equality is by the Nullstellensatz. Since $\left(0\right)$ is a prime ideal, this implies either $\sum f_iA$ or $\sum h_iA$ is the zero ideal; i.e., either $f_i$ are all zero or $h_i$ are all zero. Finally, $A$ is an inductive limit of finitely generated ''k''-algebras that are integral domains and thus, using the previous property, $A \otimes_k B = \varinjlim A_i \otimes_k B$ is an integral domain. $\square$ and, in algebraic geometry, it implies the statement that the coordinate ring of the product of two affine algebraic varieties over an algebraically closed field is again an integral domain.

# Field of fractions

The
field of fractions In abstract algebra, the field of fractions of an integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
''K'' of an integral domain ''R'' is the set of fractions ''a''/''b'' with ''a'' and ''b'' in ''R'' and ''b'' ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containing ''R'' " in the sense that there is an injective ring homomorphism such that any injective ring homomorphism from ''R'' to a field factors through ''K''. The field of fractions of the ring of integers $\Z$ is the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s $\Q.$ The field of fractions of a field 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 the field itself.

# Algebraic geometry

Integral domains are characterized by the condition that they are reduced (that is ''x''2 = 0 implies ''x'' = 0) and irreducible (that is there is only one
minimal prime 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 ...
). The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains. The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal. This translates, 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 ... , into the fact that the
coordinate ring 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 o ...
of an
affine algebraic set Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certai ...
is an integral domain if and only if the algebraic set is an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
. More generally, a commutative ring is an integral domain if and only if 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 ...
is an
integral 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 a ...
affine scheme In commutative algebra Commutative algebra is the branch of 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 wit ...
.

# Characteristic and homomorphisms

The
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
of an integral domain is either 0 or 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 ...
. If ''R'' is an integral domain of prime characteristic ''p'', then the
Frobenius endomorphism In commutative algebra and field theory (mathematics), field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative Ring (mathematics), rings with prime characteristic (algebra), characteristic ...
''f''(''x'') = ''x''''p'' is
injective 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 ...
.