B. Dhara - Academia.edu (original) (raw)

Papers by B. Dhara

Research paper thumbnail of On Lie ideals satisfying certain differential identities in prime rings

Extracta Mathematicae

Let R be a prime ring of characteristic not 2, L a nonzero square closed Lie ideal of R and let F... more Let R be a prime ring of characteristic not 2, L a nonzero square closed Lie ideal of R and let F : R → R, G : R → R be generalized derivations associated with derivations d : R → R, g : R → R respectively. In this paper, we study several conditions that imply that the Lie ideal is central. Moreover, it is shown that the assumption of primeness of R can not be removed.

Research paper thumbnail of Central Values Involving Generalized Skew Derivations and Polynomials in Prime Rings

Siberian Mathematical Journal

Research paper thumbnail of X-generalized skew derivations with annihilating and centralizing conditions in prime rings

Annali Dell'universita' Di Ferrara, Apr 6, 2022

Research paper thumbnail of Universal J. Computational Math. 2(3) 48-55, 2014

Research paper thumbnail of Generalization of generalized derivations acting as homomorphisms or anti-homomorphisms with central values on Lie ideals in prime rings

Miskolc Mathematical Notes, 2015

Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F; G;... more Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F; G; H be the generalized derivations with associated derivations d; ı; h of R respectively.

Research paper thumbnail of Vanishing Power Values of Commutators with Derivations on Prime Rings

International Journal of Mathematics and Mathematical Sciences, 2009

LetRbe a prime ring of charR≠2,da nonzero derivation ofRandρa nonzero right ideal ofRsuch that[[d... more LetRbe a prime ring of charR≠2,da nonzero derivation ofRandρa nonzero right ideal ofRsuch that[[d(x),x]n,[y,d(y)]m]t=0for allx,y∈ρ, wheren≥0,m≥0,t≥1are fixed integers. If[ρ,ρ]ρ≠0, thend(ρ)ρ=0.

Research paper thumbnail of No. 1 May2QQ5

Tamsui Oxford Journal of …, 2005

Let R be a prime ring with a derivation d and U be its nonzero Lie ideal. Ifa ER, such that a (d ... more Let R be a prime ring with a derivation d and U be its nonzero Lie ideal. Ifa ER, such that a (d (u)) mun= 0 for all u EU or aun (d (u)) m= ofor all u EU and rn, n fixed positive integers, then (i) a= 0 or d (U)= 0 if char R:: f: 2,(ii) a= 0 or d (R)= 0 if [U, U]:: f: 0 and R~ M 2 ...

Research paper thumbnail of Commutator identity involving generalized derivations on multilinear polynomials

ANNALI DELL'UNIVERSITA' DI FERRARA, 2016

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U , C be the ex... more Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U , C be the extended centroid of R, F and G be two nonzero generalized derivations of R and f (x 1 ,. .. , x n) be a multilinear polynomial over C which is not central valued on R. If [F(u)u, G(v)v] = 0 for all u, v ∈ f (R), then there exist a, b ∈ U such that F(x) = ax and G(x) = bx for all x ∈ R with [a, b] = 0 and f (x 1 ,. .. , x n) 2 is central valued on R.

Research paper thumbnail of Some theorems of commutativity on semiprime rings with mappings

Let R be a semiprime ring and F : R → R a mapping such that F (xy) = F (y)x + yd(x) for all x, y ... more Let R be a semiprime ring and F : R → R a mapping such that F (xy) = F (y)x + yd(x) for all x, y ∈ R, where d is any map on R. In this paper, we investigate the commutativity of semiprime rings with a mapping F on R. Several theorems of commutativity of semiprime rings are obtained.

Research paper thumbnail of A result concerning generalized derivations on multilinear polynomials in prime rings

Rendiconti del Circolo Matematico di Palermo Series 2

Suppose that R is a prime ring of characteristic different from 2 with Utumi quotient ring U , C ... more Suppose that R is a prime ring of characteristic different from 2 with Utumi quotient ring U , C = Z (U) the extended centroid of R and f (x 1 ,. .. , x n) a non-central multilinear polynomial over C. Further, assume that F and G are two generalized derivations of R and d is a nonzero derivation of R.

Research paper thumbnail of An identity on generalized derivations involving multilinear polynomials in prime rings

Proceedings - Mathematical Sciences

Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U , ext... more Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U , extended centroid C, f (x 1 ,. .. , xn) a multilinear polynomial over C, which is not central valued on R and d a nonzero derivation of R. By f (R), we mean the set of all evaluations of the polynomial f (x 1 ,. .. , xn) in R. In the present paper, we study b[d(u), u] + p[d(u), u]q + [d(u), u]c = 0 for all u ∈ f (R), which includes left sided, right sided as well as two-sided annihilating conditions of the set {[d(u), u] : u ∈ f (R)}. We also examine some consequences of this result related to generalized derivations and we prove that if F is a generalized derivation of R and d is a nonzero derivation of R such that F 2 ([d(u), u]) = 0 for all u ∈ f (R), then there exists a ∈ U with a 2 = 0 such that F (x) = xa for all x ∈ R or F (x) = ax for all x ∈ R.

Research paper thumbnail of A result concerning nilpotent values with generalized skew derivations on Lie ideals

Communications in Algebra

Let n ≥ 1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the ext... more Let n ≥ 1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F (x)F (y) − yx) n = 0 for all x, y ∈ L, then char(R) = 2 and R ⊆ M 2 (C), the ring of 2 × 2 matrices over C.

Research paper thumbnail of Left Annihilator of Commutator Identity with Generalized Derivations and Multilinear Polynomials in Prime Rings

Communications in Algebra

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended ... more Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F a nonzero generalized derivation of R, I an ideal of R, and f x 1 x n a multilinear polynomial over C which is not central valued on R. If 0 = a ∈ R such that a F u u F v v = 0 for all u v ∈ f I , where f I is the set of all evaluations of f x 1 x n in I, then there exists b ∈ U such that F x = bx for all x ∈ R and one of the following statements holds: (1) f x 1 x n 2 is central valued on R; (2) ab = 0.

Research paper thumbnail of Derivations vanishing on commutators with generalized derivation of order 2 in prime rings

Communications in Algebra

ABSTRACT Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and ... more ABSTRACT Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, be a multilinear polynomial over C, which is not central valued on R. If d is a nonzero derivation of R and F is a generalized derivation of R such that for all , then one of the following holds: there exists a∈U such that F(x) = xa for all x∈R with a2∈C; there exists a∈U such that F(x) = ax for all x∈R with a2∈C.

Research paper thumbnail of Multiplicative (generalized)-derivation in semiprime rings

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2015

Let R be a semiprime ring and α any mapping on R. A mapping F : R → R is called multiplicative (g... more Let R be a semiprime ring and α any mapping on R. A mapping F : R → R is called multiplicative (generalized)-derivation if F(x y) = F(x)y + xd(y) for all x, y ∈ R, where d : R → R is any map (not necessarily additive). In this paper our main motive is to study the commutativity of semiprime rings and nature of mappings.

Research paper thumbnail of Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals

Matematicki Vesnik

Let R be a prime ring with extended centroid C, L a non-central Lie ideal of R and n ≥ 1 a fixed ... more Let R be a prime ring with extended centroid C, L a non-central Lie ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that H(u 2) n = G(u) 2n for all u ∈ L, then one of the following holds: (1) H(x) = ax and G(x) = bx for all x ∈ R, with a, b ∈ C and a n = b 2n ; (2) char(R) = 2, R satisfies s 4 , H(x) = ax + [p, x] and G(x) = bx for all x ∈ R, with b ∈ C and a n = b 2n ; (3) char(R) = 2 and R satisfies s 4. As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.

Research paper thumbnail of On the Annihilators of Derivations with Engel Conditions in Prime Rings

Tamsui Oxford Journal of Mathematical Sciences, 2010

Let R be a prime ring of char R= 2, da non-zero derivation of R, 0= b∈ R and ρ a non-zero right i... more Let R be a prime ring of char R= 2, da non-zero derivation of R, 0= b∈ R and ρ a non-zero right ideal of R such that b [[d (x), x] n,[y, d (y)] m]= 0 for all x, y∈ ρ, where n, m≥ 0 are fixed integers. If [ρ, ρ] ρ= 0, then either bρ= 0 or d (ρ) ρ= 0. Keywords and Phrases: ...

Research paper thumbnail of A result on generalized derivations on Lie ideals in prime rings

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2013

Research paper thumbnail of Generalized Derivations and Commuting Additive Maps on Multilinear Polynomials in Prime Rings

Ukrainian Mathematical Journal

G(xy) = G(x)y + xd(y) holds for all x, y \in R, where d is a derivation of R. We denote [a, b] = ... more G(xy) = G(x)y + xd(y) holds for all x, y \in R, where d is a derivation of R. We denote [a, b] = ab-ba, the simple commutator of the elements a, b \in R and [a, b] k = \bigl[ [a, b] k-1 , b \bigr] , for k > 1, the kth commutator of a, b. Let T \subsete R. An additive map F : R-\rightar R is said to be commuting in T (resp. centralizing in T) if [F (x), x] = 0 for all x \in T (resp. [F (x), x] \in Z(R) for all x \in T). Several authors have studied derivations and generalized derivations which are centralizing and commuting in some subsets of prime and semiprime rings (see [12, 17, 19, 21] for references). In this view, a well-known result proved by Posner [24] states that a prime ring R must be commutative, if it admits a non-zero centralizing derivation. In [16], Lee studied derivations with Engel conditions on polynomials f (x 1 ,. .. , x n) in non-zero one-sided ideals of R. More precisely, he proved that if \bigl[ d \bigl(

Research paper thumbnail of Multiplicative (generalized)-derivations and left ideals in semiprime rings

Hacettepe Journal of Mathematics and Statistics, 2015

Research paper thumbnail of On Lie ideals satisfying certain differential identities in prime rings

Extracta Mathematicae

Let R be a prime ring of characteristic not 2, L a nonzero square closed Lie ideal of R and let F... more Let R be a prime ring of characteristic not 2, L a nonzero square closed Lie ideal of R and let F : R → R, G : R → R be generalized derivations associated with derivations d : R → R, g : R → R respectively. In this paper, we study several conditions that imply that the Lie ideal is central. Moreover, it is shown that the assumption of primeness of R can not be removed.

Research paper thumbnail of Central Values Involving Generalized Skew Derivations and Polynomials in Prime Rings

Siberian Mathematical Journal

Research paper thumbnail of X-generalized skew derivations with annihilating and centralizing conditions in prime rings

Annali Dell'universita' Di Ferrara, Apr 6, 2022

Research paper thumbnail of Universal J. Computational Math. 2(3) 48-55, 2014

Research paper thumbnail of Generalization of generalized derivations acting as homomorphisms or anti-homomorphisms with central values on Lie ideals in prime rings

Miskolc Mathematical Notes, 2015

Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F; G;... more Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F; G; H be the generalized derivations with associated derivations d; ı; h of R respectively.

Research paper thumbnail of Vanishing Power Values of Commutators with Derivations on Prime Rings

International Journal of Mathematics and Mathematical Sciences, 2009

LetRbe a prime ring of charR≠2,da nonzero derivation ofRandρa nonzero right ideal ofRsuch that[[d... more LetRbe a prime ring of charR≠2,da nonzero derivation ofRandρa nonzero right ideal ofRsuch that[[d(x),x]n,[y,d(y)]m]t=0for allx,y∈ρ, wheren≥0,m≥0,t≥1are fixed integers. If[ρ,ρ]ρ≠0, thend(ρ)ρ=0.

Research paper thumbnail of No. 1 May2QQ5

Tamsui Oxford Journal of …, 2005

Let R be a prime ring with a derivation d and U be its nonzero Lie ideal. Ifa ER, such that a (d ... more Let R be a prime ring with a derivation d and U be its nonzero Lie ideal. Ifa ER, such that a (d (u)) mun= 0 for all u EU or aun (d (u)) m= ofor all u EU and rn, n fixed positive integers, then (i) a= 0 or d (U)= 0 if char R:: f: 2,(ii) a= 0 or d (R)= 0 if [U, U]:: f: 0 and R~ M 2 ...

Research paper thumbnail of Commutator identity involving generalized derivations on multilinear polynomials

ANNALI DELL'UNIVERSITA' DI FERRARA, 2016

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U , C be the ex... more Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U , C be the extended centroid of R, F and G be two nonzero generalized derivations of R and f (x 1 ,. .. , x n) be a multilinear polynomial over C which is not central valued on R. If [F(u)u, G(v)v] = 0 for all u, v ∈ f (R), then there exist a, b ∈ U such that F(x) = ax and G(x) = bx for all x ∈ R with [a, b] = 0 and f (x 1 ,. .. , x n) 2 is central valued on R.

Research paper thumbnail of Some theorems of commutativity on semiprime rings with mappings

Let R be a semiprime ring and F : R → R a mapping such that F (xy) = F (y)x + yd(x) for all x, y ... more Let R be a semiprime ring and F : R → R a mapping such that F (xy) = F (y)x + yd(x) for all x, y ∈ R, where d is any map on R. In this paper, we investigate the commutativity of semiprime rings with a mapping F on R. Several theorems of commutativity of semiprime rings are obtained.

Research paper thumbnail of A result concerning generalized derivations on multilinear polynomials in prime rings

Rendiconti del Circolo Matematico di Palermo Series 2

Suppose that R is a prime ring of characteristic different from 2 with Utumi quotient ring U , C ... more Suppose that R is a prime ring of characteristic different from 2 with Utumi quotient ring U , C = Z (U) the extended centroid of R and f (x 1 ,. .. , x n) a non-central multilinear polynomial over C. Further, assume that F and G are two generalized derivations of R and d is a nonzero derivation of R.

Research paper thumbnail of An identity on generalized derivations involving multilinear polynomials in prime rings

Proceedings - Mathematical Sciences

Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U , ext... more Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U , extended centroid C, f (x 1 ,. .. , xn) a multilinear polynomial over C, which is not central valued on R and d a nonzero derivation of R. By f (R), we mean the set of all evaluations of the polynomial f (x 1 ,. .. , xn) in R. In the present paper, we study b[d(u), u] + p[d(u), u]q + [d(u), u]c = 0 for all u ∈ f (R), which includes left sided, right sided as well as two-sided annihilating conditions of the set {[d(u), u] : u ∈ f (R)}. We also examine some consequences of this result related to generalized derivations and we prove that if F is a generalized derivation of R and d is a nonzero derivation of R such that F 2 ([d(u), u]) = 0 for all u ∈ f (R), then there exists a ∈ U with a 2 = 0 such that F (x) = xa for all x ∈ R or F (x) = ax for all x ∈ R.

Research paper thumbnail of A result concerning nilpotent values with generalized skew derivations on Lie ideals

Communications in Algebra

Let n ≥ 1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the ext... more Let n ≥ 1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F (x)F (y) − yx) n = 0 for all x, y ∈ L, then char(R) = 2 and R ⊆ M 2 (C), the ring of 2 × 2 matrices over C.

Research paper thumbnail of Left Annihilator of Commutator Identity with Generalized Derivations and Multilinear Polynomials in Prime Rings

Communications in Algebra

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended ... more Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F a nonzero generalized derivation of R, I an ideal of R, and f x 1 x n a multilinear polynomial over C which is not central valued on R. If 0 = a ∈ R such that a F u u F v v = 0 for all u v ∈ f I , where f I is the set of all evaluations of f x 1 x n in I, then there exists b ∈ U such that F x = bx for all x ∈ R and one of the following statements holds: (1) f x 1 x n 2 is central valued on R; (2) ab = 0.

Research paper thumbnail of Derivations vanishing on commutators with generalized derivation of order 2 in prime rings

Communications in Algebra

ABSTRACT Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and ... more ABSTRACT Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, be a multilinear polynomial over C, which is not central valued on R. If d is a nonzero derivation of R and F is a generalized derivation of R such that for all , then one of the following holds: there exists a∈U such that F(x) = xa for all x∈R with a2∈C; there exists a∈U such that F(x) = ax for all x∈R with a2∈C.

Research paper thumbnail of Multiplicative (generalized)-derivation in semiprime rings

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2015

Let R be a semiprime ring and α any mapping on R. A mapping F : R → R is called multiplicative (g... more Let R be a semiprime ring and α any mapping on R. A mapping F : R → R is called multiplicative (generalized)-derivation if F(x y) = F(x)y + xd(y) for all x, y ∈ R, where d : R → R is any map (not necessarily additive). In this paper our main motive is to study the commutativity of semiprime rings and nature of mappings.

Research paper thumbnail of Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals

Matematicki Vesnik

Let R be a prime ring with extended centroid C, L a non-central Lie ideal of R and n ≥ 1 a fixed ... more Let R be a prime ring with extended centroid C, L a non-central Lie ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that H(u 2) n = G(u) 2n for all u ∈ L, then one of the following holds: (1) H(x) = ax and G(x) = bx for all x ∈ R, with a, b ∈ C and a n = b 2n ; (2) char(R) = 2, R satisfies s 4 , H(x) = ax + [p, x] and G(x) = bx for all x ∈ R, with b ∈ C and a n = b 2n ; (3) char(R) = 2 and R satisfies s 4. As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.

Research paper thumbnail of On the Annihilators of Derivations with Engel Conditions in Prime Rings

Tamsui Oxford Journal of Mathematical Sciences, 2010

Let R be a prime ring of char R= 2, da non-zero derivation of R, 0= b∈ R and ρ a non-zero right i... more Let R be a prime ring of char R= 2, da non-zero derivation of R, 0= b∈ R and ρ a non-zero right ideal of R such that b [[d (x), x] n,[y, d (y)] m]= 0 for all x, y∈ ρ, where n, m≥ 0 are fixed integers. If [ρ, ρ] ρ= 0, then either bρ= 0 or d (ρ) ρ= 0. Keywords and Phrases: ...

Research paper thumbnail of A result on generalized derivations on Lie ideals in prime rings

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2013

Research paper thumbnail of Generalized Derivations and Commuting Additive Maps on Multilinear Polynomials in Prime Rings

Ukrainian Mathematical Journal

G(xy) = G(x)y + xd(y) holds for all x, y \in R, where d is a derivation of R. We denote [a, b] = ... more G(xy) = G(x)y + xd(y) holds for all x, y \in R, where d is a derivation of R. We denote [a, b] = ab-ba, the simple commutator of the elements a, b \in R and [a, b] k = \bigl[ [a, b] k-1 , b \bigr] , for k > 1, the kth commutator of a, b. Let T \subsete R. An additive map F : R-\rightar R is said to be commuting in T (resp. centralizing in T) if [F (x), x] = 0 for all x \in T (resp. [F (x), x] \in Z(R) for all x \in T). Several authors have studied derivations and generalized derivations which are centralizing and commuting in some subsets of prime and semiprime rings (see [12, 17, 19, 21] for references). In this view, a well-known result proved by Posner [24] states that a prime ring R must be commutative, if it admits a non-zero centralizing derivation. In [16], Lee studied derivations with Engel conditions on polynomials f (x 1 ,. .. , x n) in non-zero one-sided ideals of R. More precisely, he proved that if \bigl[ d \bigl(

Research paper thumbnail of Multiplicative (generalized)-derivations and left ideals in semiprime rings

Hacettepe Journal of Mathematics and Statistics, 2015