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).

Field, commutative ring, integral domain

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

FIELD, three propositions, and their underlying definitions are presented in this white paper (knowledge base).

scholarly journals On irreducible divisor graphs in commutative rings with zero-divisors

2015 ◽  
Vol 46 (4) ◽  
pp. 365-388
Author(s):  
Christopher Park Mooney
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Zero Divisors ◽  
Classical Paper

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in integral domains. This inspired the so called irreducible divisor graph of an integral domain studied by J. Coykendall and J. Maney. Factorization in rings with zero-divisors is considerably more complicated than integral domains and has been widely studied recently. We find that many of the same techniques can be extended to rings with zero-divisors. In this article, we construct several distinct irreducible divisor graphs of a commutative ring with zero-divisors. This allows us to use graph theoretic properties to help characterize finite factorization properties of commutative rings, and conversely.

scholarly journals Vector space over a field

2021 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Vector Space ◽  
Knowledge Base ◽  
White Paper

VECTOR SPACE OVER A FIELD and its underlying definitions are presented in this white paper (knowledge base).

Supremum and infimum

2021 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Lower Bound ◽  
Knowledge Base ◽  
Upper Bound ◽  
White Paper

SUPREMUM (least upper bound), INFIMUM (greatest lower bound) 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].

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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