Integral domain - Citizendia

In abstract algebra, a branch of mathematics, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero (the zero-product property); that is, there are no zero divisors. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In the mathematical areas of algebra and analysis, the zero-product property, also known as the zero-product rule, is an abstract and explicit statement In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French An integral domain is a commutative domain. In Mathematics, especially in the area of Abstract algebra known as Ring theory, a domain is a ring with 0 &ne 1 such that ab = 0

Alternatively and equivalently, an integral domain may be defined as a commutative ring in which the zero ideal {0} is prime, or as a subring of a field. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Additionally, a commutative ring with unity R is an integral domain if and only if for every non-zero element r of the ring, the R-module map induced by multiplication by r is injective (such r are called regular). ↔

Viewing the underlying commutative ring as a preadditive category, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism, by making use of the bilinear structure on the set of morphisms). In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which

The condition 0 ≠ 1 only serves to exclude the trivial ring {0}. A trivial ring is a ring defined on a Singleton set, { r } The ring operations (× and + are trivial r \times r = r

A few sources talk about noncommutative integral domains, but we reserve the term integral domain for the commutative case and use domain for the noncommutative case. In Mathematics, especially in the area of Abstract algebra known as Ring theory, a domain is a ring with 0 &ne 1 such that ab = 0

Some specific kinds of integral domains are given with the following chain of class inclusions:

integral domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

1 Examples
2 Divisibility, prime and irreducible elements
3 Properties
4 Field of fractions
5 Algebraic geometry
6 Characteristic and homomorphisms
7 See also

Examples

The prototypical example is the ring Z of all integers. In Set theory and its applications throughout Mathematics, a subclass is a class contained in some other class in the same way that a Subset In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

Every field is an integral domain. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Conversely, every Artinian integral domain is a field. In Abstract algebra, an Artinian ring is a ring that satisfies the Descending chain condition on ideals. In particular, all finite integral domains are finite fields. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements 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:

$\mathbf{Z}\;\supset\;2\mathbf{Z}\;\supset\;\ldots\;\supset\;2^n\mathbf{Z}\;\supset\;2^{n+1}\mathbf{Z}\;\supset\;\cdots$

Rings of polynomials are integral domains if the coefficients come from an integral domain. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomials in two variables with real coefficients. In Mathematics, the real numbers may be described informally in several different ways

For each integer n > 1, the set of all real numbers of the form a + b√n with a and b integers is a subring of R and hence an integral domain. In Mathematics, the real numbers may be described informally in several different ways The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

For each integer n > 0 the set of all complex numbers of the form a + bi√n with a and b integers is a subring of C and hence an integral domain. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In the case n = 1 this integral domain is called the Gaussian integers. A Gaussian integer is a Complex number whose real and imaginary part are both Integers The Gaussian integers with ordinary addition and multiplication of complex

The p-adic integers. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897

If U is a connected open subset of the complex number plane C, then the ring H(U) consisting of all holomorphic functions f : U → C is an integral domain. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane The same is true for rings of analytic functions on connected open subsets of analytic manifolds. This article is about both real and complex analytic functions A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

If R is a commutative ring and P is an ideal in R, then the factor ring R/P is an integral domain if and only if P is a prime ideal. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers Also, R is an integral domain if and only if the ideal (0) is a prime ideal.

A regular local ring is an integral domain. In Commutative algebra, a regular local ring is a Noetherian Local ring having the property that the minimal number of generators of its Maximal ideal A deep theorem of Auslander–Buchsbaum and Nagata from the 1950s claims that, in fact, a regular local ring is a UFD. Masayoshi Nagata ( Japanese: 永田雅宜 Nagata Masayoshi; born 1927 in Aichi Japan (February 9 1927–August 27 2008 is a Japanese mathematician In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written

Divisibility, prime and irreducible elements

If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without

If a divides b and b divides c, then a divides c. If a divides b, then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference.

The elements which divide 1 are called the units of R; these are precisely the invertible elements in R. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i Units divide all other elements.

If a divides b and b divides a, then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b.

If q is a non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.

If p is a non-zero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or p divides b.

This generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 If p is a prime element, then the principal ideal (p) generated by p is a prime ideal. Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is not true in all integral domains (it is true in unique factorization domains, however). In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written

Properties

Let R be an integral domain. Then there is an integral domain S such that R ⊂ S and S has an element which is transcendental over R.
The cancellation property holds in integral domains. That is, let a, b, and c belong to an integral domain. If a ≠ 0 and ab = ac then b = c. Another way to state this is that the function x $\mapsto$ ax is injective for any non-zero a in the domain. (Recall from vector algebra that a transformation T is injective if and only if its null space consists of 0 alone. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that It is therefore possible to have a ring-isomorphic module with non-injective T — if the ring is not an integral domain. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars )

Field of fractions

If R is a given integral domain, the smallest field containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients It can be thought of as consisting of all fractions a/b with a and b in R and b ≠ 0, modulo an appropriate equivalence relation. The field of fractions of the integers is the field of rational numbers. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The field of fractions of a field is isomorphic to the field itself. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Algebraic geometry

In algebraic geometry, integral domains correspond to irreducible varieties. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given set This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety They have a unique generic point, given by the zero ideal. In Mathematics, in the fields of General topology and particularly of Algebraic geometry, a generic point P of a Topological space Integral domains are also characterized by the condition that they are reduced and irreducible. In Ring theory, a ring R is said to be reduced if it has no non- zero Nilpotent elements 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 ideal of a reduced and irreducible ring is the zero ideal, hence such rings are integral domains. The converse is clear: No integral domain can have nilpotent elements, and the zero ideal is the unique minimal ideal.

Characteristic and homomorphisms

The characteristic of every integral domain is either zero or a prime number. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

If R is an integral domain with prime characteristic p, then f(x) = x^p defines an injective ring homomorphism f : R → R, the Frobenius endomorphism. In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Commutative algebra and field theory, which are branches of Mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a

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