Ring Derivatives Research Articles

Derivation of Commutative Rings and the Leibniz Formula for Power of Derivation

Summary In this article we formalize in Mizar [1], [2] a derivation of commutative rings, its definition and some properties. The details are to be referred to [5], [7]. A derivation of a ring, say D, is defined generally as a map from a commutative ring A to A-Module M with specific conditions. However we start with simpler case, namely dom D = rng D. This allows to define a derivation in other rings such as a polynomial ring. A derivation is a map D : A → A satisfying the following conditions: (i) D(x + y) = Dx + Dy, (ii) D(xy) = xDy + yDx, ∀x, y ∈ A. Typical properties are formalized such as: D ( ∑ i = 1 n x i ) = ∑ i = 1 n D x i D\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) = \sum\limits_{i = 1}^n {D{x_i}} and D ( ∏ i = 1 n x i ) = ∑ i = 1 n x 1 x 2 ⋯ D x i ⋯ x n ( ∀ x i ∈ A ) . D\left( {\prod\limits_{i = 1}^n {{x_i}} } \right) = \sum\limits_{i = 1}^n {{x_1}{x_2} \cdots D{x_i} \cdots {x_n}} \left( {\forall {x_i} \in A} \right). We also formalized the Leibniz Formula for power of derivation D : D n ( x y ) = ∑ i = 0 n ( i n ) D i x D n - i y . {D^n}\left( {xy} \right) = \sum\limits_{i = 0}^n {\left( {_i^n} \right){D^i}x{D^{n - i}}y.} Lastly applying the definition to the polynomial ring of A and a derivation of polynomial ring was formalized. We mentioned a justification about compatibility of the derivation in this article to the same object that has treated as differentiations of polynomial functions [3].

Identities involving generalized derivations in prime rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, $$f(x_1,\ldots ,x_n)$$ be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a nonzero derivation of R and G is a generalized derivation of R. If $$G^2(u)d(u)=0$$ for all $$u\in f(R)$$ , then one of the following holds:

The twisted derivation problem for group rings

We study $$(\sigma ,\tau )$$ -derivations of a group ring RG where G is a group with center having finite index in G and R is a semiprime ring with 1 such that either R has no torsion elements or that if R has p-torsion elements, then p does not divide the order of G and let $$\sigma ,\tau $$ be R-linear endomorphisms of RG fixing the center of RG pointwise. We generalize Main Theorem 1.1 of Chaudhuri (Comm Algebra 47(9): 3800–3807, 2019) and prove that there is a ring $$T\supset R$$ such that $$\mathcal {Z}(T)\supset \mathcal {Z}(R)$$ and that for the natural extensions of $$\sigma , \tau $$ to TG, we get $$H^1(TG,{}_\sigma TG_\tau )=0$$ , where $${}_\sigma TG_\tau $$ is the twisted $$TG-TG$$ -bimodule. We provide applications of the above result and Main Theorem 1.1 of Chaudhuri (2019) to integral group rings of finite groups and connect twisted derivations of integral group rings to other important problems in the field such as the isomorphism problem and the Zassenhaus conjectures. We also give an example of a group G which is both locally finite and nilpotent and such that for every field F, there exists an F-linear $$\sigma $$ -derivation of FG which is not $$\sigma $$ -inner.

Synthesis of 4-(4-ethyl-phenyl)-3-(4-methyl-phenyl)-1,2,4-oxadiazol-5(4H)-one and 4-(4-ethyl-phenyl)-3-(4-methyl-phenyl)-1,2,4-oxadiazole-5(4H)-thione and solvent effects on their infrared spectra in organic solvents

In the present study novel 4-(4-ethyl-phenyl)-3-(4-methyl-phenyl)-1,2,4-oxadiazol-5(4H)-one (compound (4)) and 4-(4-ethyl-phenyl)-3-(4-methyl-phenyl)-1,2,4-oxadiazole-5(4H)-thione (compound (5)) were synthesized. These oxadiazole ring derivatives were characterized by IR, 1H NMR, 13C NMR and HRMS analyses. The solvent effects on CO, CN and CS stretching vibrational frequencies (ν(CO), ν(CN) and ν(CS)) of synthesized compounds were investigated experimentally using attenuated total reflection (ATR) infrared spectroscopy and compared with the theoretical results assigned using the potential energy distribution (PED) contributions. Furthermore, the ν(CO), ν(CN) and ν(CS) of compound (4) and compound (5) were correlated with empirical solvent parameters such as the solvent acceptor numbers, the Swain equation, the Kirkwood-Bauer-Magat equation, and the linear solvation energy relationships. Apart from the linear effects investigated in similar studies, solvent-induced vibrational shifts were investigated using the quadratic equation. The prediction capabilities of empirical solvent parameters were statistically compared. It was found that the linear solvation energy relationships show better correlation than the other empirical solvent parameters. Additionally, the quadratic equation provided more accurate predictions for the vibrational frequency locations than the Swain and the linear solvation energy relationships equations.

A new synthetic approach for pyrazolo[1,5-a]pyrazine-4(5H)-one derivatives and their antiproliferative effects on lung adenocarcinoma cell line.

Starting from the 3,5-dimethyl pyrazole ring and acetophenone derivatives, five different N-propargylated C-3 substituted pyrazoles were obtained. These derivatives were reacted with different amine derivatives using Cs2CO3 in methanol and 11 different pyrazolo [1,5-a] pyrazine-4(5H)-one derivatives were obtained, which are not found in the literature. The cytotoxic effects of these derivatives in the A549 cell line were investigated. The 160µM concentration of two derivatives was found to increase cell death rate to 50%, and two derivatives increased cell death rate by up to 40%. The structure-activity relationship (SAR) study revealed an amide group with a long alkyl chain and benzene ring with a p-CF3 group could be important for efficiency. With theoretical ADMET studies of pyrazolopyrazine derivatives, pharmacokinetic phases were predicted to be suitable.

A note on identities with derivations in prime rings

The present paper's primary purpose is to study the connection between commutativity of rings and the behavior of its derivations in prime rings.

A classification of generalized derivations in rings with involution

Let R be a ring. An additive mapping F : R ? R is called a generalized derivation if there exists a derivation d of R such that F(xy) = F(x)y + xd(y) for all x,y ? R. The main purpose of this paper is to characterize some specific classes of generalized derivations of rings. Precisely, we describe the structure of generalized derivations of noncommutative prime rings with involution that belong to a particular class of generalized derivations. Consequently, some recent results in this line of investigation have been extended. Moreover, some suitable examples showing that the assumed hypotheses are crucial, are also given.

Left annihilator of identities with generalized derivations in prime and semiprime rings

Left annihilator of identities with generalized derivations in prime and semiprime rings

On the skew Lie product and derivations of prime rings with involution

On the skew Lie product and derivations of prime rings with involution

Maleimide-g-pyridine Octadecene Polymers: Synthesis, Characterization, Thermal Stability, Fluorescence and Biological Studies

The fluorescent poly(maleic anhydride-alt-octadecene-arylmethoxypyridine) (PMAOA) and poly(maleic anhydride-alt-octadecene-arylchloropyridine) (PMAOB) polymers were synthesized and characterized by FTIR, 1H & 13C NMR spectroscopy. The thermal stability of polymers was measured by TGA-DTA in the range 100-700 ºC. The fluorescence properties of the synthesized PMAOA and PMAOB polymers were studied by UV-Vis spectroscopy and fluorescence spectra and showed notable report due to influence of pyridine grafted to the malelic anhydride ring derivatives. The fluorescent polymers, PMAOA and PMAOB shows high efficacy of antimicrobial activity against Candida albicans, Aspergillusniger, Staphylococcus aureus and Escherichia coli pathogens.

Organocatalytic Cascade Reactions for the Diversification of Thiopyrano-Piperidone Fused Rings Utilizing Trienamine Activation.

An aminocatalytic privileged diversity-oriented synthesis (ApDOS) strategy utilizing trienamine catalysis for the construction of diverse and complex thiopyrans-piperidone fused rings through a thia-Diels-Alder/nucleophilic ring-closing sequence by using dithioamides as activated heterodienophiles is reported. Following this strategy, a super cascade reaction to assemble nine fused rings can be achieved by employing a bis-dithioamide. Additionally, by linking an indole moiety on the dithioamide, a Pictet-Spengler type reaction can be promoted once the cascade sequence has been achieved, leading to more complex penta- hexa- and heptacyclic fused ring derivatives in a one-pot process. This investigation opens new perspectives for the synthesis of a new class of complex and diverse thiopyrans that contribute to populate new relevant regions in the chemical space.

Design, Synthesis, Molecular Docking of Novel Substituted Pyrimidinone Derivatives as Anticancer Agents

Novel derivatives of pyrimidinone ring carrying some five-membered heterocycles had been designed and prepared. 1,3-Thiazolidin-4-one derivative 7 was prepared by the reaction of 1,2,3,4-tetrahydro-4-(4-hydroxyphenyl)-6-methyl-N′-((naphthalen-4-yl)methylene)-2-oxopyrimidine-5-carbohydrazide 6 with thioglycolic acid. Reacting the hydrazide 5 with benzoyl chloride gave compound 8 which cyclized to 1,3,4-oxadiazole 9 by the use of phosphorus pentoxide. Moreover, the pyrazole derivative 10 was obtained from reacting compound 5 with ethyl acetoacetate. Furthermore, the reaction of the hydrazide 5 with phenyl isothiocyanates produced the carbothioamide 11 which refluxed with hydrazonoyl chloride to yield 1,3,4-thiadiazole derivative 14. In addition, 1, 2,4-triazole derivative 16 was prepared via reacting the carbothioamide 11 with sodium hydroxide followed by methylation with methyl iodide. The structures of the newly prepared heterocyclic compounds were determined by spectral and elemental analysis. The cytotoxic activity of the target compounds (7, 9, 10, 14, and 16) were estimated in vitro against PC-3 (prostate adenocarcinoma), human HepG-2 (liver carcinoma), and HCT116 (colorectal carcinoma) cell lines using MTT assay. All the tested compounds appeared high level of inhibition against PC-3 and HCT116 cell lines, while appeared weak inhibitory action against HepG-2 cell lines, especially, the thiadiazole derivative 14. Furthermore, docking study had been carried out on the targets 7, 9, 10, 14, and 16 within EGFR as a prediction for the anticancer activity. All the docked compounds showed good fitting with EGFR recording docking scores range (-27.99 to -19.01).

On the structure of generalized Jordan *-derivations of prime rings

Let be a noncommutative prime ring with involution and let be the maximal symmetric ring of quotients of In the present paper, we describe the structure of generalized Jordan *-derivations, i.e., additive mappings satisfying for all where d is an associated Jordan *-derivation of Precisely, we prove that under certain conditions any generalized Jordan *-derivation of is of the form for all where As a consequence, we show that any generalized Jordan *-derivation on a prime ring with involution is generalized X-inner, provided is not a PI-ring. Finally, we conclude our paper with a direction for further research.

An Engel condition with two generalized derivations in prime rings

Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that for all where are fixed positive integers. Then, one of the following conditions holds: there exist such that and for all m = s, n = t and there exist and such that g(x) = xa and for all C is a finite field, the matrix ring over C for some integer and there exist such that g(x) = xa and h(x) = ux for all

Non-cytotoxic 1,2,3-triazole tethered fused heterocyclic ring derivatives display Tax protein inhibition and impair HTLV-1 infected cells

Human T cell lymphotropic virus type 1 (HTLV-1) is a human retrovirus that infects approximately 10–20 million people worldwide and causes an aggressive neoplasia (adult T-cell leukemia/lymphoma - ATL). Therapeutic approaches for the treatment of ATL have variable effectiveness and poor prognosis, thus requiring strategies to identify novel compounds with activity on infected cells. In this sense, we initially screened a small series of 25 1,2,3-triazole derivatives to discover cell proliferation inhibitors and apoptosis inducers in HTLV-1-infected T-cell line (MT-2) for further assessment of their effect on viral tax activity through inducible-tax reporter cell line (Jurkat LTR-GFP). Eight promising compounds (02, 05, 06, 13, 15, 21, 22 and 25) with activity ≥70% were initially selected, based on a suitable cell-based assay using resazurin reduction method, and evaluated towards cell cycle, apoptosis and Tax/GFP expression analyses through flow cytometry. Compound 02 induced S phase cell cycle arrest and compounds 05, 06, 22 and 25 promoted apoptosis. Remarkably, compounds 22 and 25 also reduced GFP expression in an inducible-tax reporter cell, which suggests an effect on Tax viral protein. More importantly, compounds 02, 22 and 25 were not cytotoxic in human hepatoma cell line (Huh-7). Therefore, the discovery of 3 active and non-cytotoxic compounds against HTLV-1-infected cells can potentially contribute, as an initial promising strategy, to the development process of new drugs against ATL.

Exploration of Alternative Scaffolds for P2Y14 Receptor Antagonists Containing a Biaryl Core.

Various heteroaryl and bicyclo-aliphatic analogues of zwitterionic biaryl P2Y14 receptor (P2Y14R) antagonists were synthesized, and affinity was measured in P2Y14R-expressing Chinese hamster ovary cells by flow cytometry. Given this series' low water solubility, various polyethylene glycol derivatives of the distally binding piperidin-4-yl moiety of moderate affinity were synthesized. Rotation of previously identified 1,2,3-triazole attached to the central m-benzoic acid core (25) provided moderate affinity but not indole and benzimidazole substitution of the aryl-triazole. The corresponding P2Y14R region is predicted by homology modeling as a deep, sterically limited hydrophobic pocket, with the outward pointing piperidine moiety being the most flexible. Bicyclic-substituted piperidine ring derivatives of naphthalene antagonist 1, e.g., quinuclidine 17 (MRS4608, IC50 ≈ 20 nM at hP2Y14R/mP2Y14R), or of triazole 2, preserved affinity. Potent antagonists 1, 7a, 17, and 23 (10 mg/kg) protected in an ovalbumin/Aspergillus mouse asthma model, and PEG conjugate 12 reduced chronic pain. Thus, we expanded P2Y14R antagonist structure-activity relationship, introducing diverse physical-chemical properties.

On additive maps of MA-semirings with involution

We extend the concept of *-derivations of rings to a certain class of semirings called MA-semirings and establish some results on commutativity forced by the *-derivations satisfying different criteria. We specially focus on the results on certain conditions under which additive mappings become Jordan *-derivations.

Annihilators of power values of b-generalized derivations in prime rings

Let $R$ be a prime ring with extended centroid $C$ and maximal left ring of quotients $Q_{ml}(R)$. For a nonzero element $b\in R$ let $F:R\rightarrow R$ be a right generalized $b$-derivation associated with the map $d$ of $R$. Suppose that $s\left(F(x)\right)^n=0$ for all $x\in R$ where $s$ is a nonzero element in $R$ and $n\geq 1$ is a fixed positive integer. Then there exist some $c\in Q{ml}(R)$ and $\beta \in C$ such that $d(x)=ad_c(x)$, $F(x)=(b+\beta)xb$ for all $x\in R$ and either $s(c+\beta)=0$ or $b(c+\beta)=0$.

Multiplicative (generalized)-derivations of prime rings that act as $n$-(anti)homomorphisms

Let R be a prime ring. In this note, we describe the possible forms of multiplicative (generalized)-derivations of R that act as n-homomorphism or n-antihomomorphism on nonzero ideals of R. Consequently, from the given results one can easily deduce the results of Gusić ([7]).

Quantum-Chemistry Based Design of Halobenzene Derivatives With Augmented Affinities for the HIV-1 Viral G4/C16 Base-Pair.

The HIV-1 integrase (IN) is a major target for the design of novel anti-HIV inhibitors. Among these, three inhibitors which embody a halobenzene ring derivative (HR) in their structures are presently used in clinics. High-resolution X-ray crystallography of the complexes of the IN-viral DNA transient complex bound to each of the three inhibitors showed in all cases the HR ring to interact within a confined zone of the viral DNA, limited to the highly conserved 5′CpA 3′/5′TpG 3′ step. The extension of its extracyclic CX bond is electron-depleted, owing to the existence of the “sigma-hole.” It interacts favorably with the electron-rich rings of base G4. We have sought to increase the affinity of HR derivatives for the G4/C16 base pair. We thus designed thirteen novel derivatives and computed their Quantum Chemistry (QC) intermolecular interaction energies (ΔE) with this base-pair. Most compounds had ΔE values significantly more favorable than those of the HR of the most potent halobenzene drug presently used in clinics, Dolutegravir. This should enable the improvement in a modular piece-wise fashion, the affinities of halogenated inhibitors for viral DNA (vDNA). In view of large scale polarizable molecular dynamics simulations on the entirety of the IN-vDNA-inhibitor complexes, validations of the SIBFA polarizable method are also reported, in which the evolution of each ΔE(SIBFA) contribution is compared to its QC counterpart along this series of derivatives.