Class Of Rings Research Articles

One Weight Codes Over the Ring $F_q[v]/(v^s-1)$

In this study, we obtain one-Lee weight codes over a class of nonchain rings and study their structures.
 We give an explicit construction for one-Lee weight codes. A method to derive more one-Lee weight codes from given a one-Lee weight code is also represented. By defining and making use of a distance-preserving Gray map, we get a family of optimal one-Hamming weight codes over finite fields.

Mal'tsev--Neumann products of semi-simple classes of rings

Malt'tsev--Neumann products of semi-simple classes of associative rings are studied and some conditions which ensure that such a product is again a semi-simple class are obtained. It is shown that both products, $\mathcal{S}_{1}\circ\mathcal{S}_{2}$ and $\mathcal{S}_{2}\circ\mathcal{S}_{1}$ of semi-simple classes $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are semi-simple classes if and only if they are equal.

Bipolar Isoindoline Nitroxide for Nonaqueous Symmetrical Redox Flow Battery

AbstractCrossover in non‐aqueous redox flow batteries remains a critical challenge to the cycle stability of these devices. The use of bipolar redox‐active organic materials (ROM) is an emerging strategy for mitigating crossover. Herein we report the first example of a bipolar ROM derived from an isoindoline nitroxide, a ring class which gives a number of advantages over the more commonly employed piperidines, including greater stability and a 200 mV higher oxidation potential. Through facile synthetic transformation, the unsubstituted isoindoline nitroxide was nitrated to give a novel bipolar molecule, 5‐nitro‐1,1,3,3‐tetramethylisoindoline‐2‐yloxyl (NTMIO). This material was investigated electrochemically, revealing a reversible oxidation and quasi‐reversible reduction giving a cell potential of 2.07 V. NTMIO was then assessed as an active material in both a static and flow battery model, where cycling was observed for both oxidative and reductive redox couples for over 70 and 20 cycles respectively.

Commutative ring extensions defined by perfect-like conditions

UDC 512.5 In 2005, Enochs, Jenda, and López-Romos extended the notion of perfect rings to n -perfect rings such that a ring is n -perfect if every flat module has projective dimension less or equal than n . Later, Jhilal and Mahdou defined a commutative unital ring R to be strongly n -perfect if any R -module of flat dimension less or equal than n has a projective dimension less or equal than n . Recently Purkait defined a ring R to be n -semiperfect if R ¯ = R / R a d ( R ) is semisimple and n -potents lift modulo R a d ( R ) . We study of three classes of rings, namely, n -perfect, strongly n -perfect, and n -semiperfect rings. We investigate these notions in several ring-theoretic structures with an aim of construction of new original families of examples satisfying the indicated properties and subject to various ring-theoretic properties.

ON DISCRIMINANTS OF MINIMAL POLYNOMIALS OF THE RAMANUJAN CLASS INVARIANTS

AbstractWe study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$ . We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$ , where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$ . We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$ , where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$ . Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$ , and thus $\Delta(\mathcal{P}_n)$ , for all squarefree $n\equiv11\pmod{24}$ .

Openness of splinter loci in prime characteristic

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen–Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and F-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically étale finite extension. We also show that for a noetherian N-graded ring over a field, the homogeneous maximal ideal detects the splinter property.

Bernstein-Sato theory for singular rings in positive characteristic

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case.

The turbulence development at its initial stage: A scenario based on the idea of vortices decay

In this paper, a model of the development of a quantum turbulence in its initial stage is proposed. The origin of the turbulence in the suggested model is the decay of vortex loops with an internal structure. We consider the initial stage of this process, before an equilibrium state is established. As result of our study, the density matrix of developing turbulent flow is calculated. The quantization scheme of the classical vortex rings system is based on the approach proposed by the author earlier.

HOG1 Mitogen-Activated Protein Kinase Pathway-Related Autophagy Induced by H2O2 in Lentinula edodes Mycelia.

Mycelial ageing is associated with ROS and autophagy in Lentinula edodes. However, the underlying cellular and molecular mechanisms between ROS and autophagy remain obscure. This study induced autophagy in L. edodes mycelia through exogenous H2O2 treatment. Results showed that 100 μM H2O2 treatment for 24 h significantly inhibited mycelial growth. H2O2 caused the depolarisation of MMP and accumulation of TUNEL-positive nuclei, which was similar to the ageing phenotype of L. edodes mycelia. Transcriptome analysis showed that differentially expressed genes were enriched in the mitophagic, autophagic, and MAPK pathways. LeAtg8 and LeHog1 were selected as hub genes. RNA and protein levels of LeATG8 increased in the H2O2-treated mycelia. Using fluorescent labelling, we observed for the first time the classic ring structure of autophagosomes in a mushroom, while 3D imaging suggested that these autophagosomes surrounded the nuclei to degrade them at specific growth stages. Phospho-LeHOG1 protein can translocate from the cytoplasm to the nucleus to regulate mycelial cells, resisting ROS-induced oxidative stress. Furthermore, LeATG8 expression was suppressed when LeHOG1 phosphorylation was inhibited. These results suggest that the LeATG8-dependent autophagy in L. edodes mycelial is closely associated with the activity or even phosphorylation of LeHOG1.

On the local finite separability of finitely generated associative rings

It is proved that analogs of the theorems of M. Hall and N. S. Romanovskii are not true in the class of commutative rings. Necessary and sufficient conditions for the local finite separability of monogenic rings are established. As a corollary, it is proved that a finitely generated torsion-free PI-ring is locally finitely separable if and only if its additive group is finitely generated.

Realization of zero-divisor graphs of finite commutative rings as threshold graphs

Let R be a finite commutative ring with unity, and let $$G = (V, E)$$ be a simple graph. The zero-divisor graph, denoted by $$\Gamma (R)$$ is a simple graph with vertex set as R, and two vertices $$x, y \in R$$ are adjacent in $$\Gamma (R)$$ if and only if $$xy=0$$ . In [5], the authors have studied the Laplacian eigenvalues of the graph $$\Gamma (\mathbb {Z}_{n})$$ and for distinct proper divisors $$d_1, d_2, \cdots , d_k$$ of n, they defined the sets as, $$\mathcal {A}_{d_i} = \{x \in \mathbb {Z}_{n} : (x, n) = d_i\}$$ , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets $$\mathcal {A}_{d_i}$$ , $$1 \le i \le k$$ are actually orbits of the group action: $$Aut(\Gamma (R)) \times R \longrightarrow R$$ , where $$Aut(\Gamma (R))$$ denotes the automorphism group of $$\Gamma (R)$$ . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that $$\Gamma (R)$$ is a connected threshold graph if and only if $$R\cong {F}_{q}$$ or $$R\cong {F}_2 \times {F}_{q}$$ . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.

Atomic semigroup rings and the ascending chain condition on principal ideals

An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals stabilizes. It was asserted by P. Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by A. Grams in the seventies with a neat counterexample. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams’ first construction. In the first part of this paper, we generalize Grams’ construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct a new class of atomic semigroup rings without the ACCP.

Construction of Quantum Codes over the Class of Commutative Rings and Their Applications to DNA Codes

Let k,m be positive integers and F2m be a finite field of order 2m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring Sk=F2m[v1,v2,…,vk]⟨vi2−αivi,vivj−vjvi⟩, for i,j=1,2,3,…,k, where αi is the non-zero element of F2m. As an application, we obtain better quantum error correcting codes over the ring S1 (for k=1). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes.

The trace property in preenveloping classes

We develop the theory of trace modules up to isomorphism and explore the relationship between preenveloping classes of modules and the property of being a trace module, guided by the question of whether a given module is trace in a given preenvelope. As a consequence we identify new examples of trace ideals and trace modules, and characterize several classes of rings with a focus on the Gorenstein and regular properties.

Novel, thalidomide-like, non-cereblon binding drug tetrafluorobornylphthalimide mitigates inflammation and brain injury

BackgroundQuelling microglial-induced excessive neuroinflammation is a potential treatment strategy across neurological disorders, including traumatic brain injury (TBI), and can be achieved by thalidomide-like drugs albeit this approved drug class is compromised by potential teratogenicity. Tetrafluorobornylphthalimide (TFBP) and tetrafluoronorbornylphthalimide (TFNBP) were generated to retain the core phthalimide structure of thalidomide immunomodulatory imide drug (IMiD) class. However, the classical glutarimide ring was replaced by a bridged ring structure. TFBP/TFNBP were hence designed to retain beneficial anti-inflammatory properties of IMiDs but, importantly, hinder cereblon binding that underlies the adverse action of thalidomide-like drugs.MethodsTFBP/TFNBP were synthesized and evaluated for cereblon binding and anti-inflammatory actions in human and rodent cell cultures. Teratogenic potential was assessed in chicken embryos, and in vivo anti-inflammatory actions in rodents challenged with either lipopolysaccharide (LPS) or controlled cortical impact (CCI) moderate traumatic brain injury (TBI). Molecular modeling was performed to provide insight into drug/cereblon binding interactions.ResultsTFBP/TFNBP reduced markers of inflammation in mouse macrophage-like RAW264.7 cell cultures and in rodents challenged with LPS, lowering proinflammatory cytokines. Binding studies demonstrated minimal interaction with cereblon, with no resulting degradation of teratogenicity-associated transcription factor SALL4 or of teratogenicity in chicken embryo assays. To evaluate the biological relevance of its anti-inflammatory actions, two doses of TFBP were administered to mice at 1 and 24 h post-injury following CCI TBI. Compared to vehicle treatment, TFBP reduced TBI lesion size together with TBI-induction of an activated microglial phenotype, as evaluated by immunohistochemistry 2-weeks post-injury. Behavioral evaluations at 1- and 2-weeks post-injury demonstrated TFBP provided more rapid recovery of TBI-induced motor coordination and balance impairments, versus vehicle treated mice.ConclusionTFBP and TFNBP represent a new class of thalidomide-like IMiDs that lower proinflammatory cytokine generation but lack binding to cereblon, the main teratogenicity-associated mechanism. This aspect makes TFBP and TFNBP potentially safer than classic IMiDs for clinical use. TFBP provides a strategy to mitigate excessive neuroinflammation associated with moderate severity TBI to, thereby, improve behavioral outcome measures and warrants further investigation in neurological disorders involving a neuroinflammatory component.Graphical

The lattice coordinatized by a ring

A coordinatization functor L ( − , 1 ) : Ring → Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R ↦ L ( R ) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective ( − ) op endofunctors on Ring and Latt ; and 3) it admits localization at a left R -module. The complemented elements of L ( R , 1 ) form a partially ordered set S ( R ) isomorphic to the space of direct summands of R R . The right nonsingular rings for which the embedding of S ( R ) into the localization L ( R , 1 ) Q at the right maximal ring of quotients Q R is an isomorphism are characterized by a property that every finite matrix subgroup φ ( R R ) of the left R -module R R is essential in an element of S ( R ) . In that case, the space S ( R ) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.

Double Negative Metamaterial Based on Moebius Strip

The article proposes a new type of metamaterial cells (metacells) based on double Moebius strip. In this research, several design options have been considered which applied the variation of the metamaterial cells design and parameters. For their analysis, the Numerical modeling methods in Ansys EM Suite program were used due to the complexity of describing the interaction of metamaterial cells of non-Euclidean geometry with radio waves. Evaluation and comparison of proposed metacells have been held based on the following characteristics: the electric permittivity and the magnetic permeability as a function of frequency. Results of the researches have demonstrated the possibility of realization of double negative metamaterial (DNG) mode in the metamaterial cells based on the double Moebius strip. The strip width, its thickness, the interstrip gap, as well as the radius of the cylinder within which the metamaterial cell may be enabled, were used as mechanical parameters of the metacells. It is essential that 10-fold and more increase of the dimensions of metamaterial cell would be required for the obtainment of the same possibilities based on the classical split ring resonator (SRR). What concerns the entire scope of the potential ring options from among the considered ones, the ring with the internal area in the form of the Star of David provides the widest transmission band in the low frequency DNG mode. The corresponding area extends up to the transition frequency of 720 MHz. Moreover, the singular ring displays the invariance of the space orientation towards the presence of the DNG mode in low frequency region, when being used as the basis for the metacell.

REDUCTIONS OF IDEALS IN PRÜFER RINGS

Let R be a ring and I a proper ideal of R. An ideal J⊆I is a reduction of I if JIn=In+1 for some positive integer n; and I is called basic if it has no proper reductions. The notion of reduction was introduced by Northcott and Rees with the initial purpose to contribute to the analytic theory of ideals in Noetherian (local) rings via reductions. Two well-known results, due to Hays, assert that an integral domain is Prüfer if and only if every finitely generated ideal is basic, and it is one-dimensional Prüfer if and only if every ideal is basic. This paper investigates reductions of ideals in the family of Prüfer rings, with the aim to recover and generalize Hays’ results to classes of rings with zero-divisors subject to various Prüfer conditions.

Competing Nuclear Quantum Effects and Hydrogen-Bond Jumps in Hydrated Kaolinite.

Recent work has shown that the dynamics of hydrogen bonds in pure clays are affected by nuclear quantum fluctuations, with different effects for the hydrogen bonds holding different layers of the clay together and for those within the same layer. At the clay-water interface there is an even wider range of types of hydrogen bond, suggesting that the quantum effects may be yet more varied. We apply classical and thermostated ring polymer molecular dynamics simulations to show that nuclear quantum effects accelerate hydrogen-bond dynamics to varying degrees. By interpreting the results in terms of the extended jump model of hydrogen-bond switching, we can understand the origins of these effects in terms of changes in the quantum kinetic energy of hydrogen atoms during an exchange. We also show that the extended jump mechanism is applicable not only to the hydrogen bonds involving water, but also those internal to the clay.

On rings determined by their idempotents and units

This paper describes properties of three certain classes of rings determined by conditions on idempotents and units, namely, the condition that any two generators of each principal right ideal are associated (UG rings), the condition that every principal right ideal is generated by a sum of a unit and an idempotent (Pr ), and the condition xy = 0 implies xsy = 0 for a sum of idempotent and unit s and any elements x, y of a ring (idun-semicommutative rings). It is proved that the class of all UG rings contains every local as well as every von Neumann regular ring, and the condition Pr is satisfied by both semiperfect and regular rings. Both local and abelian regular rings are proved to be necessarily idun-semicommutative. For all three classes are presented some closure properties and illustrating examples.