In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:
S {\displaystyle S} . An integraldomain R {\displaystyle R} is said to be integrallyclosed if it is equal to its integral closure in its field of fractions...
noetherian local domain such that the integral closure is not finite over that domain.[citation needed] Let A be a noetherian integrallycloseddomain with field...
factorization domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integraldomains ⊃ integrallycloseddomains ⊃ GCD domains...
Principal ideal domains are Noetherian, they are integrallyclosed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all...
at every point, meaning that the local ring at the point is an integrallycloseddomain. An affine variety X (understood to be irreducible) is normal if...
domain. Euclidean domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integraldomains ⊃ integrallyclosed domains...
rational solution. Mathematics portal Fundamental theorem of algebra Integrallycloseddomain Descartes' rule of signs Gauss–Lucas theorem Properties of polynomial...
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...
equals its integral closure in its field of fractions is called an integrallycloseddomain. These concepts are fundamental in algebraic number theory. For...
In abstract algebra, a Schreier domain, named after Otto Schreier, is an integrallycloseddomain where every nonzero element is primal; i.e., whenever...
follows immediately that, if K is an integraldomain, then so is K[X]. It follows also that, if K is an integraldomain, a polynomial is a unit (that is,...
(I : J) is divisorial. Let R be a local Krull domain (e.g., a Noetherian integrallyclosed local domain). Then R is a discrete valuation ring if and only...
sets of the prime ideals of any commutative ring; for this topology, the closed sets are the sets of prime ideals that contain a given ideal. The spectrum...
In abstract algebra, the field of fractions of an integraldomain is the smallest field in which it can be embedded. The construction of the field of...
is an integraldomain. The lack of multiplicative inverses, which is equivalent to the fact that Z {\displaystyle \mathbb {Z} } is not closed under division...
from the so-called "quotient field", or field of fractions, of an integraldomain as well as from the more general "rings of quotients" obtained by localization...
The zero ring is generally excluded from integraldomains. Whether the zero ring is considered to be a domain at all is a matter of convention, but there...
Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain. The ring...
f−1(M) is a maximal ideal of R. If R and S are commutative and S is an integraldomain, then ker(f) is a prime ideal of R. If R and S are commutative, S is...
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