A NOTE ON DENSITY AND THE DIRICHLET CONDITION

2012 ◽  
Vol 08 (03) ◽  
pp. 823-830 ◽  
Author(s):  
FRANTIŠEK MARKO ◽  
ŠTEFAN PORUBSKÝ
Keyword(s):  
Commutative Ring ◽  
Nonzero Element ◽  
Dirichlet Condition ◽  
Zero Divisors ◽  
Open Set ◽  
Maximal Ideals ◽  
Commutative Domain

Motivated by topological approaches to Euclid and Dirichlet's theorems on infinitude of primes, we introduce and study [Formula: see text]-coprime topologies on a commutative ring R with an identity and without zero divisors. For infinite semiprimitive commutative domain R of finite character (i.e. every nonzero element of R is contained in at most finitely many maximal ideals of R), we characterize its subsets A for which the Dirichlet condition, requiring the existence of infinitely many pairwise nonassociated elements from A in every open set in the invertible topology, is satisfied.

ℎ-local Rings

2021 ◽  
pp. 93-109
Author(s):  
L. Klingler ◽  
A. Omairi
Keyword(s):  
Local Ring ◽  
Nonzero Element ◽  
Local Rings ◽  
Local Domain ◽  
Content Type ◽  
Regular Elements ◽  
Zero Divisors ◽  
Maximal Ideals ◽  
Regular Ideal ◽  
Definition Of

In the 1960’s, Matlis defined an h h -local domain to be a (commutative) integral domain in which each nonzero element is contained in only finitely many maximal ideals and each nonzero prime ideal is contained in a unique maximal ideal. For rings with zero-divisors, by changing “nonzero” to “regular,” one obtains the definition of an h h -local ring. Nearly two dozen equivalent characterizations of h h -local domain have appeared in the literature. We show that most of these remain equivalent to h h -local ring if one also replaces “localization” by “regular localization” and assumes that the ring is a Marot ring (i.e., every regular ideal is generated by its regular elements).

scholarly journals On a generalization of prime submodules of a module over a commutative ring

2017 ◽  
Vol 37 (1) ◽  
pp. 153-168
Author(s):  
Hosein Fazaeli Moghimi ◽  
Batool Zarei Jalal Abadi
Keyword(s):  
Commutative Ring ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Maximal Ideals ◽  
Prime Submodules ◽  
Prime Submodule

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.

RINGS: Almost a ring, semiring, zero, integral domain

2021 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Knowledge Base ◽  
Commutative Ring ◽  
Integral Domain ◽  
White Paper ◽  
Zero Divisors ◽  
Domain Integral ◽  
Zero Integral

RING, commutative ring, almost a ring, semiring, zero ring, zero property, zero divisors, domain, integral domain, and their underlying definitions are presented in this white paper (knowledge base).

Invariant Factors Under Rank One Perturbations

1980 ◽  
Vol 32 (1) ◽  
pp. 240-245 ◽  
Author(s):  
Robert C. Thompson
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Matrix Multiplication ◽  
Diagonal Form ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Zero Divisors ◽  
Rank One ◽  
Invariant Factors

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

Annihilating-ideal graphs of commutative rings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050121
Author(s):  
M. Aijaz ◽  
S. Pirzada
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Metric Dimension ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with unity [Formula: see text]. The annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is defined to be the graph with vertex set [Formula: see text] — the set of non-zero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] adjacent if and only if [Formula: see text]. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of [Formula: see text] in terms of their degrees as vertices of [Formula: see text]. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

scholarly journals Unique factorization in polynomial rings with zero divisors

2020 ◽  
pp. 2150113 ◽  
Author(s):  
D. D. Anderson ◽  
Ranthony A. C. Edmonds
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Unique Factorization ◽  
Factorization Property ◽  
Zero Divisors ◽  
Unique Factorization Domain

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

Modules with annihilation property

2020 ◽  
pp. 2150126
Author(s):  
Rasul Mohammadi ◽  
Ahmad Moussavi ◽  
Masoome Zahiri
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Ordered Monoid ◽  
Zero Divisors ◽  
The Right ◽  
Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

Associate class graph of zero-divisors of a commutative ring

2019 ◽  
Vol 19 (08) ◽  
pp. 2050155
Author(s):  
Gaohua Tang ◽  
Guangke Lin ◽  
Yansheng Wu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
New Class ◽  
Associate Class ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Class Graph

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring [Formula: see text], denoted by [Formula: see text]. Some properties of [Formula: see text], including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.

Cohomological operators on quotients by exact zero divisors

2020 ◽  
pp. 2250016
Author(s):  
Andrew Windle
Keyword(s):  
Commutative Ring ◽  
Zero Divisors

Let [Formula: see text] be a commutative ring, [Formula: see text] a pair of exact zero divisors, and [Formula: see text]. Let [Formula: see text] be a complex of free [Formula: see text]-modules. In this paper we explicitly compute cohomological operators of [Formula: see text] over [Formula: see text] by constructing endomorphisms of [Formula: see text]. We consider some properties of these cohomological operators, as well as provide an example in which these cohomological operators act non-trivially.

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