mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations...

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out...

elements Bézout domain, an integral domain in which the sum of two principal ideals is again a principal ideal Euclidean domain, an integral domain which allows...

said to be integral over a subring A of B if b is a root of some monic polynomial over A. If A, B are fields, then the notions of "integral over" and of...

fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero...

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 specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function...

type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more...

improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context...

In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique...

(/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain) to a function of a...

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors...

their original representations. An integral transform "maps" an equation from its original "domain" into another domain, in which manipulating and solving...

domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain...

with coefficients in an integral domain, and there are two common definitions. Most often, a polynomial over an integral domain R is said to be irreducible...

measures the "size" of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping |x| from...

z = f(x, y)) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions...

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of...

mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in...

Q(B)/Q(A).} The Noether normalization lemma implies that if R is an integral domain that is a finitely generated algebra over a field k, then the Krull...

zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideal domains, PIDs for short...

context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like...

though only one of them is positive definite. Each element of an integral domain has no more than 2 square roots. The difference of two squares identity...

follows immediately that, if K is an integral domain, then so is K[X]. It follows also that, if K is an integral domain, a polynomial is a unit (that is,...

Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which...

In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal...

In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product...

given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes...

function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains. Let R be an integral domain and g : R → Z≥0...

elements of F that are integral over R form a ring, called the integral closure of R in K. An integral domain that equals its integral closure in its field...