scholarly journals Note on a chain condition for prime ideals

1959 ◽  
Vol 32 (1) ◽  
pp. 85-90 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Chain Condition ◽  
Prime Ideals ◽  
A Chain

Some Properties of Noetherian Domains of Dimension One

1961 ◽  
Vol 13 ◽  
pp. 569-586 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Vector Space ◽  
Integral Domain ◽  
Chain Condition ◽  
Prime Ideals ◽  
Quotient Field ◽  
Rank One ◽  
Descending Chain Condition ◽  
A Chain ◽  
Dimension One ◽  
Torsion Free

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

Rings each of whose proper cyclic modules has a chain condition

2012 ◽  
pp. 28-36
Author(s):  
S. K. Jain ◽  
Ashish K. Srivastava ◽  
Askar A. Tuganbaev
Keyword(s):  
Chain Condition ◽  
A Chain

Finite p-Groups Whose Subgroups of Given Order Are Isomorphic and Minimal Non-abelian

2019 ◽  
Vol 26 (01) ◽  
pp. 1-8
Author(s):  
Qinhai Zhang
Keyword(s):  
Chain Condition ◽  
A Chain

Finite p-groups whose subgroups of given order are isomorphic and minimal non-abelian are classified. In addition, two results on a chain condition of 𝒜t-groups are improved.

scholarly journals On Reduced Zero-Divisor Graphs of Posets

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Ashish Kumar Das ◽  
Deiborlang Nongsiang
Keyword(s):  
Zero Divisor ◽  
Chain Condition ◽  
Equivalence Classes ◽  
Prime Ideals ◽  
Zero Divisors ◽  
Ascending Chain Condition

We study some properties of a graph which is constructed from the equivalence classes of nonzero zero-divisors determined by the annihilator ideals of a poset. In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.

Prime Ideals in GCD-Domains

1974 ◽  
Vol 26 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Philip B. Sheldon
Keyword(s):  
General Setting ◽  
Chain Condition ◽  
Integral Closure ◽  
Prime Ideals ◽  
Unique Factorization Domain ◽  
Principal Ideals ◽  
Valuation Rings ◽  
Complete Integral Closure ◽  
Minimal Prime ◽  
Bezout Domains

A GCD-domain is a commutative integral domain in which each pair of elements has a greatest common divisor (g.c.d.). (This is the terminology of Kaplansky [9]. Bourbaki uses the term ''anneau pseudobezoutien" [3, p. 86], while Cohn refers to such rings as "HCF-rings" [4].) The concept of a GCD-domain provides a useful generalization of that of a unique factorization domain (UFD), since several of the standard results for a UFD can be proved in this more general setting (for example, integral closure, some properties of D[X], etc.). Since the class of GCD-domains contains all of the Bezout domains, and in particular, the valuation rings, it is clear that some of the properties of a UFD do not hold in general in a GCD-domain. Among these are complete integral closure, ascending chain condition on principal ideals, and some of the important properties of minimal prime ideals.

scholarly journals A note on universally zero-divisor rings

1991 ◽  
Vol 43 (2) ◽  
pp. 233-239 ◽  
Author(s):  
S. Visweswaran
Keyword(s):  
Zero Divisor ◽  
Chain Condition ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Prüfer Domain ◽  
Integral Extension ◽  
Maximal Ideals ◽  
Unitary Module ◽  
Finite Set ◽  
Descending Chain Condition

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

$\mathcal{A}_t$-GROUPS SATISFYING A CHAIN CONDITION

2014 ◽  
Vol 13 (04) ◽  
pp. 1350137 ◽  
Author(s):  
LIHUA ZHANG ◽  
HAIPENG QU
Keyword(s):  
Positive Integer ◽  
Chain Condition ◽  
Metacyclic Group ◽  
Abelian Subgroups ◽  
A Chain

For a positive integer t, a finite p-group G is an [Formula: see text]-group if all subgroups of index pt in G are abelian, and at least one subgroup of index pt-1 in G is not abelian. An [Formula: see text]-group G satisfies a chain condition if every [Formula: see text]-subgroup of G is contained in an [Formula: see text]-subgroup for all i ∈ {0, 1, 2, …, t - 1}, where [Formula: see text]-subgroups denote abelian subgroups. We prove that if t ≥ 2, then, except for some p-groups of small order, the following conditions for a group G of order pn are equivalent: (1) G is an [Formula: see text]-group satisfying a chain condition; (2) every subgroup of order pn-k in G is an [Formula: see text]-subgroup for 0 ≤ k ≤ t; (3) G is an ordinary metacyclic group.

A note on the ascending chain condition of ideals

2019 ◽  
Vol 19 (07) ◽  
pp. 2050135 ◽  
Author(s):  
Ibrahim Al-Ayyoub ◽  
Malik Jaradat ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Regular Ideal ◽  
Reduction Number ◽  
Number Formula ◽  
A Chain ◽  
Ascending Chain Condition

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

scholarly journals An order property for families of sets

1988 ◽  
Vol 44 (3) ◽  
pp. 294-310
Author(s):  
N. H. Williams
Keyword(s):  
Pairwise Disjoint ◽  
Chain Condition ◽  
Infinite Cardinal ◽  
Order Property ◽  
Disjoint Family ◽  
Combinatorial Properties ◽  
The Family ◽  
Almost Disjoint ◽  
A Chain ◽  
Almost Disjoint Family

AbstractWe develop the idea of a θ-ordering (where θ is an infinite cardinal) for a family of infinite sets. A θ-ordering of the family A is a well ordering of A which decomposes A into a union of pairwise disjoint intervals in a special way, which facilitates certain transfinite constructions. We show that several standard combinatorial properties, for instance that of the family A having a θ-transversal, are simple consequences of A possessing a θ-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a θ-ordering. The restrictions involve either intersection conditions on A (the intersection of every λ-size subfamily of A has size at most κ) or a chain condition on A.

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