scholarly journals Note on Colon-Multiplication Domains

2010 ◽  
Vol 2010 ◽  
pp. 1-4 ◽  
Author(s):  
A. Mimouni
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Nonzero Ideal ◽  
Quotient Field ◽  
Integral Domains ◽  
Fractional Ideal ◽  
Multiplication Ideal

LetRbe an integral domain with quotient fieldL.Call a nonzero (fractional) idealAofRa colon-multiplication ideal any idealA, such thatB(A:B)=Afor every nonzero (fractional) idealBofR.In this note, we characterize integral domains for which every maximal ideal (resp., every nonzero ideal) is a colon-multiplication ideal. It turns that this notion unifies Dedekind andMTPdomains.

scholarly journals Integral ∨-ideals

1981 ◽  
Vol 22 (2) ◽  
pp. 167-172 ◽  
Author(s):  
David F. Anderson
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Dedekind Domain ◽  
Integral Ideal ◽  
Quotient Field ◽  
Fractional Ideal

Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

ON ⋆-COLON-MULTIPLICATION DOMAINS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250156
Author(s):  
OLIVIER A. HEUBO-KWEGNA
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Fractional Ideal ◽  
Star Operation ◽  
Multiplication Ideal

Let ⋆ be a star operation on an integral domain R. An ideal A is a ⋆-colon-multiplication ideal if A⋆ = (B(A : B))⋆ for all fractional ideal B of R. We prove that every maximal ideal of R is a ⋆-colon-multiplication ideal if and only if R is a ⋆-CICD or R is a local ⋆-MTP domain. It is also shown that every ideal of R is ⋆-colon-multiplication if and only if R is a ⋆-CICD.

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.

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.

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.

GENERALLY t-LINKATIVE DOMAINS

2012 ◽  
Vol 11 (02) ◽  
pp. 1250027
Author(s):  
THOMAS G. LUCAS
Keyword(s):  
Finite Type ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Type System ◽  
Nonzero Ideal ◽  
Two Dimensional ◽  
Finitely Generated ◽  
One Dimensional ◽  
Valuation Domains ◽  
Prüfer Domains

An overring t of an integral domain R is t-linked over R if for each finitely generated nonzero ideal I of R, (T : IT) ⊋ T implies (R : I) ⊋ R. A t-linkative domain is one for which each overring is t-linked. The notion of a generally t-linkative domain is introduced as a domain R such that [Formula: see text] is t-linkative for each finite type system of ideals [Formula: see text]. In general, R is generally t-linkative if and only if RM is generally t-linkative for each maximal ideal M. All Prüfer domains are generally t-linkative as are all one-dimensional domains and all pseudo-valuation domains. If R is Noetherian and not a field, then it is generally t-linkative if and only if it is one-dimensional. In contrast, an example is given of a two-dimensional Mori domain that is generally t-linkative.

On the Ideal Equation I(B ∩ C) = IB∩IC

1983 ◽  
Vol 26 (3) ◽  
pp. 331-332 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Integral Domain ◽  
Nonzero Ideal ◽  
Quotient Field ◽  
The Ideal

AbstractLet R be an integral domain with quotient field K and let I be a nonzero ideal of R. We show (1) that I is invertible if and only if for every nonempty collection {Bα} of ideals of R and (2) that I is flat if and only if I(B ∩ C) = IB∩IC for each pair of ideals B and C of R.

Integral Domains Whose w-Integral Closures Are Krull Domains

2020 ◽  
Vol 27 (02) ◽  
pp. 287-298
Author(s):  
Gyu Whan Chang ◽  
HwanKoo Kim
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Krull Domain ◽  
Integral Closures ◽  
Krull Domains

Let D be an integral domain with quotient field K, [Formula: see text] be the integral closure of D in K, and D[w] be the w-integral closure of D in K; so [Formula: see text], and equality holds when D is Noetherian or dim(D) = 1. The Mori–Nagata theorem states that if D is Noetherian, then [Formula: see text] is a Krull domain; it has also been investigated when [Formula: see text] is a Dedekind domain. We study integral domains D such that D[w] is a Krull domain. We also provide an example of an integral domain D such that [Formula: see text], t-dim(D) = 1, [Formula: see text] is a Prüfer v-multiplication domain with t-dim([Formula: see text]) = 2, and D[w] is a UFD.

ON INTEGRAL DOMAINS WITH CYCLIC GROUP ACTIONS

2011 ◽  
Vol 10 (03) ◽  
pp. 491-508
Author(s):  
JUNRO SATO ◽  
SUSUMU ODA ◽  
KEN-ICHI YOSHIDA
Keyword(s):  
New York ◽  
Cyclic Group ◽  
Galois Extension ◽  
Integral Domain ◽  
Group Actions ◽  
Local Rings ◽  
Quotient Field ◽  
Galois Extensions ◽  
Integral Domains ◽  
Field L

Let A be a commutative integral domain with quotient field L, and let R be a subdomain of A with quotient field K. Assuming that L is a Galois extension of K, Nagata required the condition for R to be normal when A is called a Galois extension of R (see p. 31, M. Nagata, Local Rings (Wiley, New York, 1962)). However in this paper, A is considered in the case that R is not necessarily assumed to be normal. We introduce the notion of cyclic Galois extensions of integral domains and investigate several properties of such ring extensions. In particular, we completely determine the seminormalization [Formula: see text] of A in an overdomain B such that both A ⊆B are cyclic Galois extensions of R.

VALUATIVE DOMAINS

2010 ◽  
Vol 09 (01) ◽  
pp. 43-72 ◽  
Author(s):  
PAUL-JEAN CAHEN ◽  
DAVID E. DOBBS ◽  
THOMAS G. LUCAS
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Nonzero Element ◽  
Prime Ideals ◽  
Quotient Field ◽  
Prüfer Domain ◽  
One Dimensional ◽  
Maximal Ideals ◽  
Valuation Domains ◽  
Integrally Closed

A (commutative integral) domain R is said to be valuative if, for each nonzero element u in the quotient field of R, at least one of R ⊆ R[u] and R ⊆ R[u-1] has no proper intermediate rings. Such domains are closely related to valuation domains. If R is a valuative domain, then R has at most three maximal ideals, and at most two if R is not integrally closed. Also, if R is valuative, the set of nonmaximal prime ideals of R is linearly ordered, at most one maximal ideal of R does not contain each nonmaximal prime of R, and RP is a valuation domain for each prime P except for at most one maximal ideal. Any integrally closed valuative domain is a Bézout domain. Valuation domains are characterized as the quasilocal integrally closed valuative domains. Each one-dimensional Prüfer domain with at most three maximal ideals is valuative.

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