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.

On a Condition of J. Ohm for Integral Domains

1968 ◽  
Vol 20 ◽  
pp. 970-983 ◽  
Author(s):  
Robert Gilmer
Keyword(s):  
Integral Domain ◽  
Closed Domain ◽  
Prüfer Domain ◽  
Integral Domains ◽  
Integrally Closed

This paper originated mainly from results presented in a paper by J. Ohm (13), and, to a lesser degree, from results of Gilmer in (3). Ohm's paper is concerned with the validity of the equation (x, y)n = (xn, yn) for each pair of elements x, y of an integral domain D with identity. If D is a Prüfer domain, the above equation is valid for all x, y ϵ D (7, p. 244). Butts and Smith have shown (2) that if (x, y)2 = (x2, y2) for all x, y of the integrally closed domain D, then D is a Prüfer domain.

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.

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 Nearly integral homomorphisms of commutative rings

1989 ◽  
Vol 40 (1) ◽  
pp. 1-12 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Integral Domain ◽  
Commutative Rings ◽  
Integral Closure ◽  
Prüfer Domain

A unital homomorphism f: R → T of commutative rings is said to be nearly integral if the induced map R/I → T/IT is integral for each ideal I of R which properly contains ker (f). This concept leads to new characterisations of integral extensions and fields. For instance, if R is not a field, then an inclusion R → T is integral if and only if it is nearly integral and (R, T) is a lying-over pair. It is also proved that each overring extension of an integral domain R is nearly integral if and only if dim (R) ≤ 1 and the integral closure of R is a Prüfer domain. Related properties and examples are also studied.

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).

scholarly journals Note on integral closures of semigroup rings

2000 ◽  
Vol 31 (2) ◽  
pp. 137-144
Author(s):  
Ryuki Matsuda
Keyword(s):  
Commutative Ring ◽  
Quotient Group ◽  
Integral Domain ◽  
Regular Ring ◽  
Quotient Ring ◽  
Closed Domain ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Integrally Closed

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

Pairs of domains where most of the intermediate domains are Prüfer

2020 ◽  
pp. 2150101
Author(s):  
Noômen jarboui
Keyword(s):  
Integral Domain ◽  
Prüfer Domain ◽  
Integral Domains ◽  
Integrally Closed ◽  
Ring Extensions

Let [Formula: see text] be an extension of integral domains. The ring [Formula: see text] is said to be maximal non-Prüfer subring of [Formula: see text] if [Formula: see text] is not a Prüfer domain, while each subring of [Formula: see text] properly containing [Formula: see text] is a Prüfer domain. Jaballah has characterized this kind of ring extensions in case [Formula: see text] is a field [A. Jaballah, Maximal non-Prüfer and maximal non-integrally closed subrings of a field, J. Algebra Appl. 11(5) (2012) 1250041, 18 pp.]. The aim of this paper is to deal with the case where [Formula: see text] is any integral domain which is not necessarily a field. Several examples are provided to illustrate our theory.

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.

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