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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, 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.
In mathematics, more specifically in abstract algebra, the concept of integrally closed has three meanings: A commutative ring R {\displaystyle R} contained in a commutative ring S {\displaystyle S} is said to be integrally closed in S {\displaystyle S} if R {\displaystyle R} is equal to the integral closure of R {\displaystyle R} in S ...
Any UFD is integrally closed. In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial with coefficients in R, then k is an element of R. Let S be a multiplicatively closed subset of a UFD A. Then the localization S −1 A is a UFD. A partial converse to this also holds; see below.
Furthermore, if A is an integrally closed domain, then the going-down holds (see below). In general, the going-up implies the lying-over. [7] Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over". When A, B are domains such that B is integral over A, A is a field if and only if B is a field.
All principal ideal domains are integrally closed. The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain. Let A be an integral domain, the following are equivalent. A is a PID. Every prime ideal of A is principal. [13] A is a Dedekind domain that is a UFD.
Radical ideals (e.g., prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed. In a normal ring, for any non-zerodivisor x and any ideal I, ¯ = ¯. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. [1] [2] Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility.
The ring = of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered ...