In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. In other words, a Dedekind domain is a commutative ring which is not a field, doesn't have zero divisors, and in which every ideal is finitely generated, every nonzero prime ideal is a maximal ideal, and which is integrally closed in its fraction field. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ... In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... 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 ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

An alternative characterization of Dedekind domains is that an integral domain R is a Dedekind domain if and only if the localization of R at each prime ideal P of R is a discrete valuation ring. In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... In mathematics, a discrete valuation ring (DVR) is a particular kind of commutative ring that is a local ring, which satisfies conditions that in algebraic geometry come from non-singularity of a point on an algebraic curve. ...

Some examples of Dedekind domains are the ring of integers, the polynomial ring F[X] in one variable over any field F, and any other principal ideal domain. Not all Dedekind domains are principal ideal domains however. The most important examples of Dedekind domains, and historically the motivating ones, arise from algebraic number fields: start with a finite field extension F of the rational numbers Q and consider the set of all elements of F which are algebraic integers (in other words, the integral closure of Z in F). This is a Dedekind domain, and F is its fraction field. A concrete example is the set { ai√2 + b√2 + ci + d : a, b, c, d in Z }, considered as a subring of C. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers. ...

The study of Dedekind domains began when Dedekind introduced the notion of ideal in a ring in the hopes of compensating for the failure of unique factorization into primes in rings of algebraic integers. While not all Dedekind domains are unique factorization domains, they all have the following property which is in practice often "close enough": every ideal can be uniquely factored as a product of prime ideals. This explains why Dedekind thought of ideals as "idealized numbers". Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ... 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. ...

If we think of ideals as whole numbers, then the fractional ideals play the role of fractions. If R is a Dedekind domain with fraction field E, then a fractional ideal I is an additive subgroup of E such that RI ⊆ I and such that there exists an r in R with rI ⊆ R. These fractional ideals can be added and multiplied like ordinary ideals, and the non-zero ones can be inverted: I^-1 := {x in E : xI ⊆ R}. It is then true that II^-1 = R. The unique factorization from above extends to fractional ideals: any fractional ideal can be uniquely written as a product of prime ideals of R and their inverses. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

A Dedekind domain is a unique factorization domain if and only if it is a principal ideal domain. The ideal class group measures the failure of unique factorization in a Dedekind domain (by measuring the failure of ideals to be principal). In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ...