DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS

Dhara, Basudeb,Kar, Sukhendu,Mondal, Sachhidananda Korean Mathematical Society 2013 대한수학회보 Vol.50 No.5

Let R be a prime ring, I a nonzero ideal of R, $d$ a derivation of R, $m({\geq}1)$, $n({\geq}1)$ two fixed integers and $a{\in}R$. (i) If $a((d(x)y+xd(y)+d(y)x+yd(x))^n-(xy+yx))^m=0$ for all $x,y{\in}I$, then either $a=0$ or R is commutative; (ii) If $char(R){\neq}2$ and $a((d(x)y+xd(y)+d(y)x+yd(x))^n-(xy+yx)){\in}Z(R)$ for all $x,y{\in}I$, then either $a=0$ or R is commutative.

NOTES ON GENERALIZED DERIVATIONS ON LIE IDEALS IN PRIME RINGS

Dhara, Basudeb,Filippis, Vincenzo De Korean Mathematical Society 2009 대한수학회보 Vol.46 No.3

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that $u^sH(u)u^t$ = 0 for all u $\in$ L, where s $\geq$ 0, t $\geq$ 0 are fixed integers. Then H(x) = 0 for all x $\in$ R unless char R = 2 and R satisfies $S_4$, the standard identity in four variables.

Curcuma Dye with Modified Treatment Using Chitosan for Superior Fastness

Dhara Shukla,Padma S Vankar 한국섬유공학회 2018 Fibers and polymers Vol.19 No.9

Curcuma dye could never gain popularity in natural dyeing as the intrinsic nature of the colorant Curcumin did not have the requisite adhering property towards natural fabrics such as silk and cotton, thereby making it very fugitive. In this paper attempts have been made to activate Curcumin molecule by complexation with chitosan (another natural linear polysaccharide). The binding took place at intrinsic pH (7-8) very effectively without any surfactant or enzyme. Dyeing with this new composite showed excellent wash and light fastnesses as compared to curcuma dye. The novelty of this dyeing process is the remarkable enhancement of wash and light fastnesses levels by 1.0-1.5 for silk and cotton fabric dyeing in just 1 hour at 40-45 oC. It is fast and energy conserving dyeing process. Three subsequent washing of the dyed samples showed very small change in CIE Lab.

b-GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS IN PRIME RINGS

Dhara, Basudeb Korean Mathematical Society 2018 대한수학회보 Vol.55 No.2

Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

Notes on generalized derivations on Lie ideals in prime rings

Basudeb Dhara,Vincenzo De Filippis 대한수학회 2009 대한수학회보 Vol.46 No.3

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that u^sH(u)u^t=0 for all u ∈ L, where s≥ 0, t≥0 are fixed integers. Then H(x)=$ for all x∈ R unless char R=2 and R satisfies S_(4), the standard identity in four variables. Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that u^sH(u)u^t=0 for all u ∈ L, where s≥ 0, t≥0 are fixed integers. Then H(x)=$ for all x∈ R unless char R=2 and R satisfies S_(4), the standard identity in four variables.

Antifungal Textile Dyeing with Mahonia napaulensis D.C. Leaves Extract Based on Its Antifungal Activity

Bajpai, Dhara,Vankar, Padma S The Korean Fiber Society 2007 Fibers and polymers Vol.8 No.5

The antifungal activity of the methanolic extract of Mahonia napaulensis D.C. or Taming (local name) leaves was evaluated with four species of common pathogenic fungi e.g. Colletotrichum capsici (MTCC No. 2071), Leptosphaerulin trifoli (MTCC No. 2328), Alternaria brassicicola (MTCC No. 2102) and Helminthosporium solani (MTCC No. 2075). The present investigation also aims at developing an eco friendly natural antifungal finish from plant extract of Mahonia for textile application. The antifungal textile dyeing was also carried out with aqueous extract of stem and leaves of Mahonia and the dyed fabric was tested against fungal species Trichoderma for its antifungal activity in vitro. Mahonia extract showed substantial antifungal activity of 83.33 % for Leptosphaerulin trifoli and Alternaria brassicicola by 80 ppm dose in 24 hours and 46 % antifungal activity in Mahonia dyed pieces in broth against Trichoderma.

$b$-generalized derivations on multilinear polynomials in prime rings

Basudeb Dhara 대한수학회 2018 대한수학회보 Vol.55 No.2

Let $R$ be a noncommutative prime ring of characteristic different from $2$, $Q$ be its maximal right ring of quotients and $C$ be its extended centroid. Suppose that $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C$, $b\in Q$, $F$ a $b$-generalized derivation of $R$ and $d$ is a nonzero derivation of $R$ such that $$d([F(f(r)),f(r)])=0$$ for all $r=(r_1,\ldots,r_n)\in R^n$. Then one of the following holds: (1) there exists $\lambda\in C$ such that $F(x)=\lambda x$ for all $x\in R$; (2) there exist $\lambda\in C$ and $p\in Q$ such that $F(x)=\lambda x+px+xp$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued in $R$.

Derivations with annihilator conditions in prime rings

Basudeb Dhara,Sukhendu Kar,Sachhidananda Mondal 대한수학회 2013 대한수학회보 Vol.50 No.5

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R, m(≥ 1), n(≥ 1) two fixed integers and a ∈ R. (i) If a((d(x)y + xd(y) + d(y)x + yd(x))n − (xy + yx))m = 0 for all x, y ∈ I, then either a = 0 or R is commutative; (ii) If char(R) 6≠2 and a((d(x)y + xd(y) + d(y)x +yd(x))n −(xy +yx)) ∈ Z(R) for all x, y ∈ I, then either a = 0 or R is commutative.

GENERALIZED DERIVATIONS WITH CENTRALIZING CONDITIONS IN PRIME RINGS

Das, Priyadwip,Dhara, Basudeb,Kar, Sukhendu Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.1

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R and f($x_1,{\ldots},x_n$) a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all the evaluations of f($x_1,{\ldots},x_n$) on R. If d is a nonzero derivation of R and G a nonzero generalized derivation of R such that $$d(G(u)u){\in}Z(R)$$ for all $u{\in}f(R)$, then $f(x_1,{\ldots},x_n)^2$ is central-valued on R and there exists $b{\in}U$ such that G(x) = bx for all $x{\in}R$ with $d(b){\in}C$. As an application of this result, we investigate the commutator $[F(u)u,G(v)v]{\in}Z(R)$ for all $u,v{\in}f(R)$, where F and G are two nonzero generalized derivations of R.

Olax psittacorum (Lam.) Vahl. (Olacaceae): current status and future prospects as an herbal plant

Majumder Raja,Adhikari Lopamudra,Dhara Moonmun 경희대학교 융합한의과학연구소 2022 Oriental Pharmacy and Experimental Medicine Vol.22 No.2

Olax psittacorum (Lam.) Vahl. traditionally used by the tribal community of ‘INDIA’ to heal conditions such as pain, psoriasis, mouth-ulcer, anemia, constipation as well as diabetes. Documented scientific shreds of evidence like antipyretic, anti-inflammatory, antimicrobial, anti-viral, anti-cancer, toxicity profile improved its medicinal value. Chromatographic evaluation (GC–MS) proven near about presence of 25 compounds. Among them, hexadecanoic methyl ester, β-sitosterol, phytol, methyl salicylate, 13-docosenamide, and oleanolic acid are significant and contribute to its successful medicinal properties. The toxicity study also revealed the safety of the consumption of the leaf part. Still, consecutive gaps need to be exploring to gather evidence of its contribution to curing different ailments, especially upon the ulcer, anemia, and identification of the presence of lead compounds as an upcoming future research proposal. The ‘Internet Scholar’, ‘Study Guide’, ‘PubMed’, ‘ScienceDirect’, ‘Scopus’, and ‘Sodhganga’ web site have been accessed to gather information about detailed scientific research as well as traditional information from authenticated sources that have been published in the English language. The various folklore claimed medicinal properties like anti-ulcer, aphthous ulcer, analgesic, anti-diabetic, anti-anemic, etc. yet to be justified scientifically with lead compounds that can support the rationality being used as folklore. This plant could be a good choice to explore its importance within the pharmacognostic field of drug development and might be a better source of herbal derived lead compounds which can help to heal ulcer or anemia that can effectively make a credible addition to treatment among the civilization as a natural source of therapeutic compounds.