In algebra, the integral closure of an ideal I of a commutative ring R, denoted by I ¯ {\displaystyle {\overline {I}}} , is the set of all elements r in...

\mathbb {N} } -graded subring of B. There is also a concept of the integral closure of an ideal. The integral closure of an ideal I ⊂ R {\displaystyle I\subset...

closures. The most basic one is the radical of an ideal. Another is the integral closure of an ideal. Given an irredundant primary decomposition I = ∩ Q...

results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient...

{\overline {J^{j}}}\subset J^{n_{j}}} , where the bar means the integral closure of an ideal. The second says that a Noetherian local domain is analytically...

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,...

The integral closure of an integral domain in a field that contains it. The radical of an ideal in a commutative ring. In geometry, the convex hull of a...

theorem holds for integral closure of an ideal). If K is a field, then the ring R = K[x,y]/(x2,xy) (the coordinate ring of a line with an embedded point)...

The absolute integral closure is the integral closure of an integral domain in an algebraic closure of the field of fractions of the domain. absolutely...

Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure. It is named after Richard Dedekind who...

{1}{r}}(I\cap (r))} (as long as R is an integral domain) The above properties can be used to calculate the quotient of ideals in a polynomial ring given their...

after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a...

product of fields Tensor product of R-algebras Quotient ring Field of fractions Product of rings Annihilator (ring theory) Integral closure Completion...

In commutative algebra, an N-1 ring is an integral domain A {\displaystyle A} whose integral closure in its quotient field is a finitely generated A {\displaystyle...

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

Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind...

is an integral domain or a local ring, it holds that ⋂ n > 0 a n = 0 {\displaystyle \bigcap _{n>0}{{\mathfrak {a}}^{n}}=0} for any proper ideal 𝔞 of R...

Eisenbud-Harris, The geometry of schemes. Springer-Verlag, 197, 2000 Swanson, Irena; Huneke, Craig (2006). Integral Closure of Ideals, Rings, and Modules. Cambridge...

algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x−1 belongs to D....

coefficients of the equations of the variety. In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal. As the...

rank n of OK as a free Z-module is equal to the degree of K over Q. A useful tool for computing the integral closure of the ring of integers in an algebraic...

the ring of integers of K {\displaystyle K} is an integral domain. The field K {\displaystyle K} is the field of fractions of the integral domain O K...

theorem, which characterizes a Cohen–Macaulay ring, holds for integral closure of an ideal; specifically, for a Noetherian ring A {\displaystyle A} , the...

normalization of an irreducible affine variety is affine; the coordinate ring of the normalization is the integral closure of the coordinate ring of the variety...

and integral closure are intimately connected to multiplicity theory. Let R be a positively graded ring such that R is finitely generated as an R0-algebra...

fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique...

{\displaystyle C} of the polynomials with coefficients in K {\displaystyle K} form an algebraically closed field called an algebraic closure of K . {\displaystyle...

normal ring. Huneke, Craig; Swanson, Irena (2006), "Ch. 19", Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note...

{\displaystyle R} is an integral domain, K {\displaystyle K} its field of fractions, and S {\displaystyle S} is the integral closure of R {\displaystyle R}...

Semiprime ring, Reduced ring Integral domain, Domain (ring theory) Field of fractions, Integral closure Euclidean domain, Principal ideal domain, Unique factorization...