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...

In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a commutative ring without nonzero zero divisors) in which every ideal...

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...

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...

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...

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

In mathematics, a GCD domain (sometimes called just domain) is an integral domain R with the property that any two elements have a greatest common divisor...

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...

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...

(/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...

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...

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...

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,...

In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is also a principal ideal. This means that Bézout's identity...

ring is an integral domain, the set of the fractions form a field. The field F(x) of the rational fractions over a field (or an integral domain) F is the...

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...

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...

absolute value. More precisely, if D is a field or (more generally) an integral domain, an absolute value on D is a function, commonly denoted | x | , {\displaystyle...

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

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...

their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring. Algebraic geometry...

thus: "to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning...

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ euclidean domains ⊃ fields...

In mathematics, an overring of an integral domain contains the integral domain, and the integral domain's field of fractions contains the overring. Overrings...

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...

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...

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...

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ euclidean domains ⊃ fields...

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...