Construction of the integral closure of a finite integral domain

1970 ◽  
Vol 40 (1) ◽  
pp. 101-120 ◽  
Author(s):  
Abraham Seidenberg
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Finite Integral

Condensed Domains

2003 ◽  
Vol 46 (1) ◽  
pp. 3-13 ◽  
Author(s):  
D. D. Anderson ◽  
Tiberiu Dumitrescu
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Field Extension ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Closed Domain ◽  
Local Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Finite Integral

AbstractAn integral domain D with identity is condensed (resp., strongly condensed) if for each pair of ideals I, J of D, IJ = {ij ; i ∈ I; j ∈ J} (resp., IJ = iJ for some i ∈ I or IJ = Ij for some j ∈ J). We show that for a Noetherian domain D, D is condensed if and only if Pic(D) = 0 and D is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain D is strongly condensed if and only if D is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension k ⊆ K, the domain D = k + XK[[X]] is condensed if and only if [K : k] ≤ 2 or [K : k] = 3 and each degree-two polynomial in k[X] splits over k, while D is strongly condensed if and only if [K : k] ≤ 2.

scholarly journals On complete integral closure and Archimedean valuation domains

1996 ◽  
Vol 61 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Robert Gilmer
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Extension Field ◽  
Quotient Field ◽  
Valuation Domains ◽  
Complete Integral Closure

AbstractSuppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

Ordinary Singularities with Decreasing Hilbert Function

1982 ◽  
Vol 34 (1) ◽  
pp. 169-180 ◽  
Author(s):  
Leslie G. Roberts
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Hilbert Function ◽  
Integral Closure ◽  
Minimal Prime Ideal ◽  
Quotient Field ◽  
Graded Ring ◽  
Image Position ◽  
Minimal Prime ◽  
Dimension One

Let A be the co-ordinate ring of a reduced curve over a field k. This means that A is an algebra of finite type over k, A has no nilpotent elements, and that if P is a minimal prime ideal of A, then A/P is an integral domain of Krull dimension one. Let M be a maximal ideal of A. Then G(A) (the graded ring of A relative to M) is defined to be . We get the same graded ring if we first localize at M, and then form the graded ring of AM relative to the maximal ideal MAM. That isLet Ā be the integral closure of A. If P1, P2, …, Ps are the minimal primes of A thenwhere A/Pi is a domain and is the integral closure of A/Pi in its quotient field.

FINITELY VALUATIVE DOMAINS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250112 ◽  
Author(s):  
PAUL-JEAN CAHEN ◽  
DAVID E. DOBBS ◽  
THOMAS G. LUCAS
Keyword(s):  
Nonnegative Integer ◽  
Integral Domain ◽  
Nonzero Element ◽  
Integral Closure ◽  
Quotient Field ◽  
Prüfer Domain ◽  
Maximal Ideals ◽  
Saturated Chain

For a pair of rings S ⊆ T and a nonnegative integer n, an element t ∈ T\S is said to be within n steps of S if there is a saturated chain of rings S = S0 ⊊ S1 ⊊ ⋯ ⊊ Sm = S[t] with length m ≤ n. An integral domain R is said to be n-valuative (respectively, finitely valuative) if for each nonzero element u in its quotient field, at least one of u and u-1 is within n (respectively, finitely many) steps of R. The integral closure of a finitely valuative domain is a Prüfer domain. Moreover, an n-valuative domain has at most 2n + 1 maximal ideals; and an n-valuative domain with 2n + 1 maximal ideals must be a Prüfer domain.

An answer to a problem about the number of overrings

2016 ◽  
Vol 15 (06) ◽  
pp. 1650022 ◽  
Author(s):  
M. Ben Nasr
Keyword(s):  
Open Problem ◽  
Integral Domain ◽  
Integral Closure ◽  
Closed Domain ◽  
Prüfer Domain ◽  
Finite Spectrum ◽  
Integrally Closed

Let [Formula: see text] be an integral domain with only finitely many overrings, equivalently, a domain such that its integral closure [Formula: see text] is a Prüfer domain with finite spectrum and there are only finitely many rings between [Formula: see text] and [Formula: see text]. Jaballah solved the problem of counting the overrings in the case [Formula: see text] but left the general case as an open problem [A. Jaballah, The number of overrings of an integrally closed domain, Expo. Math. 23 (2005) 353–360, Problem 3.4]. The purpose of this paper is to provide a solution to that problem.

WHEN IS THE INTEGRAL CLOSURE COMPARABLE TO ALL INTERMEDIATE RINGS

2016 ◽  
Vol 95 (1) ◽  
pp. 14-21 ◽  
Author(s):  
MABROUK BEN NASR ◽  
NABIL ZEIDI
Keyword(s):  
Integral Domain ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Necessary And Sufficient Conditions ◽  
Integral Domains ◽  
Necessary And Sufficient ◽  
Quotient Rings

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$. We study the set of intermediate rings between $R$ and $S$. We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

scholarly journals Perinormalitas di Daerah Krull (Perinormality on Krull Domains)

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Qonita Qurrota A'yun ◽  
Sri Wahyuni
Keyword(s):  
Prime Ideal ◽  
Integral Domain ◽  
Integral Closure ◽  
Normal Domain ◽  
Closedness Property ◽  
Krull Domains

Daerah integral R dikatakan perinormal jika untuk setiap overring (lokal) T dari R yang memenuhi kondisi going-down, maka T merupakan lokalisasi dari R pada ideal prima. Perinormalitas merupakan salah satu sifat ketertutupan integral. Dengan memperhatikan bahwa klosur integral dari daerah normal Noether merupakan daerah Krull, akan ditunjukkan bagaimana sifat perinormalitas di daerah Krull.An integral domain R is said to be perinormal if whenever T is a (local) overring of R such that the inclusion R in T satisfies going-down, it follows that T is a localization of R necessarily at a prime ideal. Perinormality is one of integral closedness property. As the integral closure of any Noetherian normal domain is Krull, it will be shown how perinormality behaves on Krull domains.

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