scholarly journals A note on certain subsets of algebraic integers

1969 ◽  
Vol 1 (3) ◽  
pp. 345-352
Author(s):  
T.W. Atterton
Keyword(s):  
Integral Domain ◽  
Finite Extension ◽  
Integral Closure ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Inclusion Relations ◽  
Integrally Closed

This paper is concerned with certain subsets of a finite extension K of the quotient field of an integral domain R. These subsets are contained in the integral closure of R in K and when R is integrally closed they are identical with it, but generally they need not even be rings. Various inclusion relations are studied and examples are given to show that these inclusions may be strict (with one exception which is still undecided).

On Integral Closure

1954 ◽  
Vol 6 ◽  
pp. 471-473 ◽  
Author(s):  
Hubert Butts ◽  
Marshall Hall ◽  
H. B. Mann
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Monic Polynomial ◽  
Unit Element ◽  
Integral Closure ◽  
Quotient Field ◽  
Integrally Closed

Let J be an integral domain (i.e., a commutative ring without divisors of zero) with unit element, F its quotient field and J[x] the integral domain of polynomials with coefficients from J . The domain J is called integrally closed if every root of a monic polynomial over J which is in F also is in J.

scholarly journals Some Remarks on Complete Integral Closure

1969 ◽  
Vol 9 (3-4) ◽  
pp. 310-314 ◽  
Author(s):  
William Heinzer
Keyword(s):  
Positive Integer ◽  
Integral Domain ◽  
Nonzero Element ◽  
Integral Closure ◽  
Quotient Field ◽  
Integrally Closed ◽  
Complete Integral Closure

This paper continues an investigation of the complete integral closure of an integral domain which was begun in [2]. We recall that if D is an integral domain with quotient field K then an element x of K is said to be almost integral over D if there exists a nonzero element y of D such that yxn is an element of D for each positive integer n. The set D* of elements of K almost integral over D is called the complete integral closure of D and D is said to be completely integrally closed if D* = D.

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.

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.

When is R[θ] integrally closed?

2016 ◽  
Vol 15 (05) ◽  
pp. 1650091 ◽  
Author(s):  
Sudesh K. Khanduja ◽  
Bablesh Jhorar
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Minimal Polynomial ◽  
Closed Domain ◽  
Necessary And Sufficient Condition ◽  
Quotient Field ◽  
Maximal Ideals ◽  
Dedekind Domains ◽  
Integrally Closed ◽  
Necessary And Sufficient

Let [Formula: see text] be an integrally closed domain with quotient field [Formula: see text] and [Formula: see text] be an element of an integral domain containing [Formula: see text] with [Formula: see text] integral over [Formula: see text]. Let [Formula: see text] be the minimal polynomial of [Formula: see text] over [Formula: see text] and [Formula: see text] be a maximal ideal of [Formula: see text]. Kummer proved that if [Formula: see text] is an integrally closed domain, then the maximal ideals of [Formula: see text] which lie over [Formula: see text] can be explicitly determined from the irreducible factors of [Formula: see text] modulo [Formula: see text]. In 1878, Dedekind gave a criterion known as Dedekind Criterion to be satisfied by [Formula: see text] for [Formula: see text] to be integrally closed in case [Formula: see text] is the localization [Formula: see text] of [Formula: see text] at a nonzero prime ideal [Formula: see text] of [Formula: see text]. Indeed he proved that if [Formula: see text] is the factorization of [Formula: see text] into irreducible polynomials modulo [Formula: see text] with [Formula: see text] monic, then [Formula: see text] is integrally closed if and only if for each [Formula: see text], either [Formula: see text] or [Formula: see text] does not divide [Formula: see text] modulo [Formula: see text], where [Formula: see text]. In 2006, a similar necessary and sufficient condition was given by Ershov for [Formula: see text] to be integrally closed when [Formula: see text] is the valuation ring of a Krull valuation of arbitrary rank (see [Comm. Algebra. 38 (2010) 684–696]). In this paper, we deal with the above problem for more general rings besides giving some equivalent versions of Dedekind Criterion. The well-known result of Uchida in this direction proved for Dedekind domains has also been deduced (cf. [Osaka J. Math. 14 (1977) 155–157]).

scholarly journals A Note on Quotient Fields of Power Series Rings

1994 ◽  
Vol 37 (2) ◽  
pp. 162-164 ◽  
Author(s):  
Huah Chu ◽  
Yi-Chuan Lang
Keyword(s):  
Power Series ◽  
Integral Domain ◽  
Algebraic Degree ◽  
Closed Domain ◽  
Quotient Field ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Power Series Rings

AbstractLet R be an integral domain with quotient field K. If R has an overling S ≠ K, such that S[X] is integrally closed, then the "algebraic degree" of K((X)) over the quotient field of R[X] is infinite. In particular, it holds for completely integrally closed domain or Noetherian domain R.

scholarly journals Krull modules and completely integrally closed modules

2021 ◽  
Author(s):  
Mu’amar Musa Nurwigantara ◽  
Indah Emilia Wijayanti ◽  
Hidetoshi Marubayashi ◽  
Sri Wahyuni
Keyword(s):  
Integral Domain ◽  
Free Module ◽  
Minimal Prime Ideal ◽  
Closed Formula ◽  
Multiplication Module ◽  
Quotient Field ◽  
Integrally Closed ◽  
Torsion Free Module ◽  
Minimal Prime ◽  
Torsion Free

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

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