scholarly journals Classification of chain rings

2021 ◽  
Vol 7 (4) ◽  
pp. 5106-5116
Author(s):  
Yousef Alkhamees ◽  
◽  
Sami Alabiad
Keyword(s):  
Artinian Ring ◽  
Chain Ring ◽  
A Chain

<abstract><p>An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.</p></abstract>

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

2014 ◽  
Vol 21 (02) ◽  
pp. 249-256 ◽  
Author(s):  
G. Aalipour ◽  
S. Akbari ◽  
M. Behboodi ◽  
R. Nikandish ◽  
M. J. Nikmehr ◽  
...  
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Vertex Set

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

Structural systematics of 4,4′-disubstituted benzenesulfonamidobenzenes. 1. Overview and dimer-based isostructures

2007 ◽  
Vol 63 (4) ◽  
pp. 621-632 ◽  
Author(s):  
Thomas Gelbrich ◽  
Michael B. Hursthouse ◽  
Terence L. Threlfall
Keyword(s):  
Hydrogen Bonding ◽  
Crystal Structures ◽  
Triple Bond ◽  
Common Type ◽  
Dimensional Structure ◽  
One Dimensional ◽  
Distinct Structure ◽  
Polymorphic Forms ◽  
A Chain

One hundred 4,4′-disubstituted benzenesulfonamidobenzenes, X–C6H5–SO2–NH–C6H5–Y, where X, Y = NO2, CN, CF3, I, Br, Cl, F, H, Me, OMe, have been synthesized and their crystal structures determined. The resulting set of 133 structures, which includes polymorphic forms, is used to make a comparative study of the molecular packing and the nature of the intermolecular interactions, including the formation of hydrogen-bonding motifs and the influence of the two substituents X and Y on these features. Nine distinct supramolecular connectivity motifs of hydrogen bonding are encountered. There are 74% of all the structures investigated which exhibit one of two motifs based on N—H...O=S interactions, a dimer or a chain. There are three other, infrequent motifs, also employing N—H...O=S links, which exhibit more complexity. Four different chain motifs result from either N—H...O=N, N—H...C[triple-bond]N or N—H...OMe interactions, arising from the presence of a nitro (position Y), nitrile (X or Y) or methoxy (Y) substituent. The program XPac [Gelbrich & Hursthouse (2005). CrystEngComm, 7, 324–336] was used to systematically analyse the packing relationships between crystal structures. Similar discrete (zero-dimensional) and extended (one-dimensional and two-dimensional) structure components, as well as cases of isostructurality were identified. A hierarchy for the classification of the 56 distinct structure types of this set is presented. The most common type, a series of 22 isostructures containing the simple centrosymmetric N—H...O=S-bonded dimer, is discussed in detail.

scholarly journals Official recognition of facts as an institution of international law: problems and

2020 ◽  
pp. 222-238
Author(s):  
Kira L'vovna Sazonova
Keyword(s):  
Intergovernmental Relations ◽  
Practical Interest ◽  
The State ◽  
The Status ◽  
The Subject ◽  
The Russian Federation ◽  
The Common ◽  
A Chain ◽  
Official Recognition

We are witnessing a formation of the new institution of recognition, which can be referred to as the &ldquo;official recognition of facts&rdquo;. Such seemingly different political themes as annexation of Crimea by the Russian Federation, the &ldquo;Skripal Case&rdquo;, or the status of the Golan Heights have an important common parameter &ndash; each of them has become an object of recognition by at least one country. Examination of the causal links that conduce certain countries to issuing the acts of recognition of long-past events or territorial changes are of considerable scientific and practical interest. Recognition of facts by the state is of paramount importance, as it[WU1]&nbsp; is documented and reflects stance on a specific event, fact, or occurrence. Recognition ensures legitimacy for further actions of the state and initiates a chain of related political and legal events, including sanctions. Over the recent years, recognition of facts by the countries has become more frequent, and virtually becomes a means of political manipulation. Classification of the facts and events that have most often been the subject of recognition allows determining the common trends in the procedure of recognition, as well as the factors that prompt the country to resort to such step. Thus, at times strange and illogical actions of the state associated with the official recognition or non-recognition of the fact acquire a specific political and legal meaning, and allow analyzing the new strategic vectors in intergovernmental relations. &nbsp;[WU1]

scholarly journals Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

2015 ◽  
Vol 15 (02) ◽  
pp. 1650029 ◽  
Author(s):  
Leandro Cagliero ◽  
Fernando Szechtman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Weight Space ◽  
Arbitrary Field ◽  
Nilpotent Radical ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
A Chain ◽  
Algebra Over A Field

Let 𝔤 be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical 𝔯 and nilpotent radical 𝔫 = [𝔤, 𝔯]. Given a finite-dimensional 𝔤-module U, its nilpotency series 0 ⊂ U(1) ⊂ ⋯ ⊂ U(m) = U is defined so that U(1) is the 0-weight space of 𝔫 in U, U(2)/U(1) is the 0-weight space of 𝔫 in U/U(1), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial 𝔤/𝔫-module, i.e. its 𝔤/𝔫-submodules form a chain. Every uniserial 𝔤-module is linked, every linked 𝔤-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked 𝔤-modules when 𝔤 = 〈x〉 ⋉ 𝔞 and ad x acts diagonalizably on the abelian Lie algebra 𝔞. Moreover, we identify and classify all uniserial 𝔤-modules amongst them.

Automorphisms of a chain ring

2006 ◽  
Vol 186 (2) ◽  
pp. 289-301
Author(s):  
Surjeet Singh ◽  
Yousef Alkhamees
Keyword(s):  
Chain Ring ◽  
A Chain

Non-free extensions of the simplex codes over a chain ring with four elements

2012 ◽  
Vol 66 (1-3) ◽  
pp. 27-38 ◽  
Author(s):  
Thomas Honold ◽  
Ivan Landjev
Keyword(s):  
Chain Ring ◽  
Simplex Codes ◽  
A Chain

Quantum codes from cyclic codes over F3 + vF3

2014 ◽  
Vol 12 (06) ◽  
pp. 1450042 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Commutative Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Quantum Code ◽  
Chain Ring ◽  
Arbitrary Length ◽  
Orthogonal Codes ◽  
Finite Commutative Ring ◽  
A Chain

Let R = F3 + vF3 be a finite commutative ring, where v2 = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F3 as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

scholarly journals Asymmetric Cryptosystem on Matrix Algebra Over a Chain Ring

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 45
Author(s):  
Muzna Yumman ◽  
Tariq Shah ◽  
Iqtadar Hussain
Keyword(s):  
Matrix Algebra ◽  
Discrete Logarithm ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Fundamental Change ◽  
Chain Ring ◽  
Asymmetric Cryptography ◽  
Novel Approach ◽  
A Chain ◽  
Asymmetric Cryptosystem

The revolutionary idea of asymmetric cryptography brings a fundamental change to our modern communication system. However, advances in quantum computers endanger the security of many asymmetric cryptosystems based on the hardness of factoring and discrete logarithm, while the complexity of the quantum algorithm makes it hard to implement in many applications. In this respect, novel asymmetric cryptosystems based on matrices over residue rings are in practice. In this article, a novel approach is introduced. [...]

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