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.

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.

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.

Graded integral domains in which each nonzero homogeneous t-ideal is divisorial

2019 ◽  
Vol 18 (01) ◽  
pp. 1950018 ◽  
Author(s):  
Gyu Whan Chang ◽  
Haleh Hamdi ◽  
Parviz Sahandi
Keyword(s):  
Integral Domain ◽  
Nonzero Ideal ◽  
Integral Domains ◽  
Cancellative Monoid ◽  
Integrally Closed ◽  
Graded Integral Domain

Let [Formula: see text] be a nonzero commutative cancellative monoid (written additively), [Formula: see text] be a [Formula: see text]-graded integral domain with [Formula: see text] for all [Formula: see text], and [Formula: see text]. In this paper, we study graded integral domains in which each nonzero homogeneous [Formula: see text]-ideal (respectively, homogeneous [Formula: see text]-ideal) is divisorial. Among other things, we show that if [Formula: see text] is integrally closed, then [Formula: see text] is a P[Formula: see text]MD in which each nonzero homogeneous [Formula: see text]-ideal is divisorial if and only if each nonzero ideal of [Formula: see text] is divisorial, if and only if each nonzero homogeneous [Formula: see text]-ideal of [Formula: see text] is divisorial.

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]).

WHEN ARE GRADED INTEGRAL DOMAINS ALMOST GCD-DOMAINS

2013 ◽  
Vol 23 (08) ◽  
pp. 1909-1923 ◽  
Author(s):  
JUNG WOOK LIM
Keyword(s):  
Integral Domain ◽  
Integral Domains ◽  
Closed Set ◽  
Integrally Closed ◽  
Graded Integral Domain

Let R = ⨁α∈Γ Rα be a (Γ-)graded integral domain and let H be the multiplicatively closed set of nonzero homogeneous elements of R. In this paper, we introduce the concepts of graded almost GCD-domains (graded AGCD-domain) and graded almost Prüfer v-multiplication domains (graded AP v MD ). Among other things, we show that if R is integrally closed, then (1) H is an almost lcm splitting set of R if and only if R is a graded AGCD-domain and (2) R is a graded AP v MD if and only if R is a P v MD . We also give an example of a (non-integrally closed) graded AGCD-domain (respectively, graded AP v MD ) that is not an almost GCD-domain (respectively, almost Prüfer v-multiplication domain.

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.

INTEGRAL DOMAINS OF THE FORM A + XI[[X]]: PRIME SPECTRUM, KRULL DIMENSION

2005 ◽  
Vol 04 (06) ◽  
pp. 599-611
Author(s):  
SANA HIZEM ◽  
ALI BENHISSI
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Integral Domain ◽  
Finite Dimension ◽  
Krull Dimension ◽  
Necessary Condition ◽  
Prime Spectrum ◽  
Prüfer Domain ◽  
Integral Domains ◽  
Contraction Map

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring [Formula: see text] First, we study the prime spectrum of R. We prove that the contraction map: Spec (A[[X]]) → Spec (R); Q ↦ Q ∩ R induces a homeomorphism, for the Zariski's topologies, from {Q ∈ Spec (A[[X]]) | XI[[X]] ⊈ Q} onto {P ∈ Spec (R) | XI[[X]] ⊈ P}. If P ∈ Spec (R) is such that XI[[X]] ⊆ P then there exists p ∈ Spec (A) such that P = p + XI[[X]]. Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim (R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim (R) = dim (A[[X]]) = dim (A) + 1. Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.

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).

On the Ideal Theory of the Kronecker Function Ring and the Domain D(X)

1969 ◽  
Vol 21 ◽  
pp. 558-563 ◽  
Author(s):  
Jimmy T. Arnold
Keyword(s):  
Ideal Theory ◽  
Dedekind Domain ◽  
Closed Domain ◽  
Quotient Field ◽  
Prüfer Domain ◽  
Function Ring ◽  
Integrally Closed ◽  
The Ideal

Let D be an integrally closed domain with identity having quotient field L. If {Vα} is the set of valuation overrings of D and if A is an ideal of D, then à = ∪ αAVα is an ideal of D called the completion of A. If X is an indeterminate over D and f ∈ D[X], then we denote by Af the ideal of D generated by the coefficients of f. The Kronecker function ring DK of D is defined by DK = {f/g| f, g ∈ D[X], Ãf ⊆ Ag} (4, p. 558); and the domain D(X) is defined by D(X) = {f/g| f, g ∈ D[X], Ag = D} (5, p. 17). In this paper we wish to relate the ideal theory of D to that of DK and D(X) for the case in which D is a Prüfer domain, a Dedekind domain, or an almost Dedekind domain.

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