On Chain Conditions in Integral Domains

1984 ◽  
Vol 27 (3) ◽  
pp. 351-359 ◽  
Author(s):  
Valentina Barucci ◽  
David E. Dobbs
Keyword(s):  
Integral Domain ◽  
Chain Condition ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Integral Domains ◽  
Chain Conditions ◽  
Ascending Chain Condition

AbstractThe following two theorems are proved. If R is an Archimedean conducive integral domain, then R is quasilocal and dim(R) ≤1. If each overring of an integral domain R has ascending chain condition on divisorial ideals, then the integral closure of R is a Dedekind domain. Both theorems sharpen results already known in the Noetherian case. The second theorem leads to a strengthened converse of the Krull-Akizuki Theorem. We also investigate the effect of restricting the hypothesis in the second theorem to the proper overrings 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.

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].

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.

Banach algebras satisfying certain chain conditions on closed ideals

2020 ◽  
Vol 57 (3) ◽  
pp. 290-297
Author(s):  
Abdullah Alahmari ◽  
Falih A. Aldosray ◽  
Mohamed Mabrouk
Keyword(s):  
Banach Algebra ◽  
Jacobson Radical ◽  
Banach Algebras ◽  
Chain Condition ◽  
Chain Conditions ◽  
Finite Dimensional ◽  
Descending Chain Condition ◽  
Unital Banach Algebra ◽  
Ascending Chain Condition

AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

Nil Subrings of Goldie Rings are Nilpotent

1969 ◽  
Vol 21 ◽  
pp. 904-907 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Chain Condition ◽  
Chain Conditions ◽  
Direct Sum ◽  
Left Annihilator ◽  
Ascending Chain Condition

Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring.Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly.Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings.Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.

scholarly journals The asymptotic and integral closure operations in multiplicative lattice modules

2005 ◽  
Vol 36 (4) ◽  
pp. 345-358
Author(s):  
Sylvia M. Foster ◽  
Johnny A. Johnson
Keyword(s):  
Chain Condition ◽  
Integral Closure ◽  
Multiplicative Lattice ◽  
Closure Operations ◽  
Ascending Chain Condition

This paper is primarily concerned with the integral and asymptotic closure operations on a multiplicative lattice relative to the greatest element of a lattice module having the ascending chain condition. We show that a cancellation law holds for the asymptotic closure of elements of the multiplicative lattice and we ultimately show, by means of multiplicative filtrations and filtration transforms, that the asymptotic closure of an element in a multiplicative lattice relative to the greatest element of a lattice module, coincides with its integral closure relative to this element in the lattice module.

Chain Conditions for Modular Lattices with Finite Group Actions

1979 ◽  
Vol 31 (3) ◽  
pp. 558-564 ◽  
Author(s):  
Joe W. Fisher
Keyword(s):  
Modular Lattice ◽  
Chain Condition ◽  
Common Fixed Points ◽  
Finite Subgroup ◽  
Modular Lattices ◽  
Finite Group Actions ◽  
Chain Conditions ◽  
Combinatorial Result ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

scholarly journals Definitions of integral elements and quotient rings over non-commutative rings with identity

1972 ◽  
Vol 13 (4) ◽  
pp. 433-446 ◽  
Author(s):  
T. W. Atterton
Keyword(s):  
Associative Ring ◽  
Chain Condition ◽  
Integral Closure ◽  
Commutative Case ◽  
P 75 ◽  
Integrally Closed ◽  
Complete Integral Closure ◽  
Image Position ◽  
Quotient Rings ◽  
Ascending Chain Condition

Let B be an associative ring with identity, A a subring of B containing the identity of B. If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a1, a2, …, an A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A-module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If Ā = A, A is said to be completely integrally closed in B. More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B. If B is not commutative, the set A of elements of B almost integral over A, may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.

Primary Ideals in Prüfer Domains

1966 ◽  
Vol 18 ◽  
pp. 1024-1030 ◽  
Author(s):  
Jack Ohm
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
Quotient Ring ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Prüfer Domain ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Primary Ideals

A Prüfer domain is an integral domainDwith the property that for every proper prime idealPofDthe quotient ringDPis a valuation ring. Examples of such domains are valuation rings and Dedekind domains, a Dedekind domain being merely a noetherian Prüfer domain. The integral closure of the integers in an infinite algebraic extension of the rationals is another example of a Prüfer domain (5, p. 555, Theorem 8). This third example has been studied initially by Krull (4) and then by Nakano (8).

INTEGRAL AND COMPLETE INTEGRAL CLOSURES OF IDEALS IN INTEGRAL DOMAINS

2011 ◽  
Vol 10 (04) ◽  
pp. 701-710
Author(s):  
A. MIMOUNI
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Integral Domains ◽  
Dedekind Domains ◽  
Integrally Closed ◽  
Complete Integral Closure ◽  
Integral Closures ◽  
The Ideal

This paper studies the integral and complete integral closures of an ideal in an integral domain. By definition, the integral closure of an ideal I of a domain R is the ideal given by I′ ≔ {x ∈ R | x satisfies an equation of the form xr + a1xr-1 + ⋯ + ar = 0, where ai ∈ Ii for each i ∈ {1, …, r}}, and the complete integral closure of I is the ideal Ī ≔ {x ∈ R | there exists 0 ≠ = c ∈ R such that cxn ∈ In for all n ≥ 1}. An ideal I is said to be integrally closed or complete (respectively, completely integrally closed) if I = I′ (respectively, I = Ī). We investigate the integral and complete integral closures of ideals in many different classes of integral domains and we give a new characterization of almost Dedekind domains via the complete integral closure of ideals.

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