scholarly journals N-th root rings

1987 ◽  
Vol 35 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Henry Heatherly ◽  
Altha Blanchet
Keyword(s):  
Commutative Ring ◽  
Subdirect Product ◽  
General Construction ◽  
Fixed Integer ◽  
Chain Conditions

A ring for which there is a fixed integer n ≥ 2 such that every element in the ring has an n-th in the ring is called an n-th root ring. This paper gives numerous examples of diverse types of n-th root rings, some via general construction procedures. It is shown that every commutative ring can be embedded in a commutative n-th root ring with unity. The structure of n-th root rings with chain conditions is developed and finite n-th root rings are completely classified. Subdirect product representations are given for several classes of n-th root rings.

scholarly journals Modules with Chain Conditions on S-Closed Submodules

2017 ◽  
Vol 30 (3) ◽  
pp. 227
Author(s):  
Rana Noori Majeed Mohammed
Keyword(s):  
Commutative Ring ◽  
Essential Extension ◽  
Chain Conditions ◽  
Closed Submodule ◽  
Closed Submodules

  Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called  s- closed submodule denoted by  D ≤sc W, if D has   no  proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In  this  paper,  we study  modules which satisfies  the ascending chain  conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.

scholarly journals Some chain conditions on weak incidence algebras

2005 ◽  
Vol 2005 (15) ◽  
pp. 2389-2397
Author(s):  
Surjeet Singh ◽  
Fawzi Al-Thukair
Keyword(s):  
Commutative Ring ◽  
Partially Ordered Set ◽  
Nonempty Subset ◽  
Ordered Set ◽  
Partially Ordered ◽  
Incidence Algebra ◽  
Chain Conditions ◽  
Incidence Algebras

LetXbe any partially ordered set,Rany commutative ring, andT=I∗(X,R)the weak incidence algebra ofXoverR. LetZbe a finite nonempty subset ofX,L(Z)={x∈X:x≤z   for some   z∈Z}, andM=Tez. Various chain conditions onMare investigated. The results so proved are used to construct some classes of right perfect rings that are not left perfect.

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 655-672
Author(s):  
K. Selvakumar ◽  
M. Subajini
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Zero Divisor ◽  
Necessary Conditions ◽  
Commutative Rings ◽  
Nonzero Ideal ◽  
Fixed Integer ◽  
Zero Divisors ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

Centralizers and Vanishing Derivations on Multi-linear Polynomials in Prime Rings

2009 ◽  
Vol 16 (01) ◽  
pp. 23-36
Author(s):  
V. De Filippis
Keyword(s):  
Commutative Ring ◽  
Prime Rings ◽  
Fixed Integer ◽  
Prime Algebra ◽  
Standard Identity ◽  
Linear Polynomial ◽  
Linear Polynomials

Let R be a prime algebra over a commutative ring K with characteristic not equal to 2. Let d and δ be non-zero derivations of R, f(x1,…, xn) a multi-linear polynomial over K with n non-commuting variables, and m ≥ 1 a fixed integer. We prove that if δ (d(f(r1,…, rn))m) = 0 for any r1,…, rn ∈ R, then either f(x1,…, xn) is central valued on R or R satisfies the standard identity s4.

On the genus of the k-maximal hypergraph of commutative rings

2019 ◽  
Vol 11 (01) ◽  
pp. 1950010
Author(s):  
K. Selvakumar ◽  
V. C. Amritha
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Local Rings ◽  
Maximal Elements ◽  
Fixed Integer ◽  
Isomorphism Classes ◽  
Vertex Set ◽  
Non Local ◽  
Genus One

Let [Formula: see text] be a commutative ring with identity and [Formula: see text], a fixed integer. Let [Formula: see text] be the set of all [Formula: see text]-maximal elements in [Formula: see text] Associate a [Formula: see text]-maximal hypergraph [Formula: see text] to [Formula: see text] with vertex set [Formula: see text] and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for all [Formula: see text]. In this paper, we determine all isomorphism classes of finite commutative non-local rings with identity whose [Formula: see text]-maximal hypergraph has genus one. Finally, we classify all finite commutative non-local rings [Formula: see text] for which [Formula: see text] is projective.

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

A Transferable, Polarisable Force Field for Ionic Liquids

2019 ◽  
Author(s):  
Kateryna Goloviznina ◽  
José N. Canongia Lopes ◽  
Margarida Costa Gomes ◽  
Agilio Padua
Keyword(s):  
Ionic Liquids ◽  
Force Field ◽  
Force Fields ◽  
Dipole Moments ◽  
Van Der Waals Interactions ◽  
First Principles Calculations ◽  
Ion Diffusion ◽  
Fixed Integer ◽  
Molecular Compounds ◽  
Induced Dipoles

A general, transferable polarisable force field for molecular simulation of ionic liquids and their mixtures with molecular compounds is developed. This polarisable model is derived from the widely used CL\&P fixed-charge force field that describes most families of ionic liquids, in a form compatible with OPLS-AA, one of the major force fields for organic compounds. Models for ionic liquids with fixed, integer ionic charges lead to pathologically slow dynamics, a problem that is corrected when polarisation effects are included explicitly. In the model proposed here, Drude induced dipoles are used with parameters determined from atomic polarisabilities. The CL\&P force field is modified upon inclusion of the Drude dipoles, to avoid double-counting of polarisation effects. This modification is based on first-principles calculations of the dispersion and induction contributions to the van der Waals interactions, using symmetry-adapted perturbation theory (SAPT) for a set of dimers composed of positive, negative and neutral fragments representative of a wide variety of ionic liquids. The fragment approach provides transferability, allowing the representation of a multitude of cation and anion families, including different functional groups, without need to re-parametrise. Because SAPT calculations are expensive an alternative predictive scheme was devised, requiring only molecular properties with a clear physical meaning, namely dipole moments and atomic polarisabilities. The new polarisable force field, CL\&Pol, describes a broad set set of ionic liquids and their mixtures with molecular compounds, and is validated by comparisons with experimental data on density, ion diffusion coefficients and viscosity. The approaches proposed here can also be applied to the conversion of other fixed-charged force fields into polarisable versions.<br>

ФІГУРА УСЛАВЛЕННЯ ЯК ТЕМБРАЛЬНЕ НАПОВНЕННЯ ПАРТІЇ КНЯЗЯ ІГОРЯ В ОДНОЙМЕННІЙ ОПЕРІ О. БОРОДІНА

2020 ◽  
Author(s):  
Admink Admink
Keyword(s):  
Cultural Studies ◽  
General Construction ◽  
Modern Sense

Продемонстровано уявлення в сучасному українському мистецтвознавстві та культурології відомостей про тембральну стратегію й загальну конструкцію опери «Князь Ігор» за авторськими установками у науковому передбаченні сучасного смислу розуміння подій «Слова о полку Ігоревім» в композиції твору О. Бородіна.Ключові слова: славлення, тембр співу, опера, семантика баса, «Князь Ігор» О. Бородіна. The article demonstrates the presentation in Ukrainian art and cultural studies of information about the timbre strategy and the general construction of the opera Prince Igor in accordance with the author's attitudes and scientific foresight of the modern sense of understanding the events of the Word of Igor's Campaign in the composition of A. Borodin.Key words: glorification, timbre of singing, opera, bass semantics, «Prince Igor» by A. Borodin

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