Basudeb Dhara

• M.Sc, Ph. D
• Professor (Associate) at Belda College

Let \(\mathcal {R}\) be a prime ring with \(char(\mathcal {R})\ne 2\), \(\mathcal {U}\) the Utumi quotient ring of \(\mathcal {R}\) and \(\mathcal {C}=\mathcal {Z}(\mathcal {U})\) the extended centroid of \(\mathcal {R}\) and \(\mathcal {L}\) a non-central Lie ideal of \(\mathcal {R}\). If \(\mathcal {F}\), \(\mathcal {G}\) and \(\mathcal {H}\) are...

Let $R$ be a prime ring and $L$ a nonzero square closed Lie ideal of $R$. Suppose $F,G,H:R\rightarrow R$ are three multiplicative (generalized)-derivations associated with the maps $\delta,g, h:R\rightarrow R$ respectively which are not necessarily additive or derivations. Assume that $E,T:R\to R$ be any two maps (not necessarily additive). Let $d...

Let $R$ be any prime ring of char $(R)\neq 2,3$, $L$ a non-central Lie ideal of $R$ and $F$, $G$ two non-zero $b$-generalized skew derivations of $R$. Suppose that $[F(u),u][G(u),u]=0$ for all $u\in L$, then following one conclusion holds: \begin{enumerate} \item $F(x)=\lambda x$ for all $x\in R$ and for some $\lambda \in C$, where $C$ is the exte...

Let $\mathfrak{R}$ be a prime ring of characteristic different from $2$, $\mathcal{Q}_r^m$ be its maximal right ring of quotients, $\mathcal{C}$ be its extended centroid and $\omega(s_1,\ldots,s_n)$ be a noncentral multilinear polynomial over $\mathcal{C}$. Suppose that $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are three $X$-generalized...

In this paper we are going to show that derivations satisfying some identity carry a certain form. To prove this, we assume \(\mathcal {R}\) is a prime ring with \(char(\mathcal {R})\ne 2\), \(\mathcal {I}\) is a nonzero ideal of \(\mathcal {R}\), \(\mathcal {U}\) is the Utumi quotient ring of \(\mathcal {R}\) with extended centroid \(\mathcal {C}=...

Let R be a prime ring of char (R) \neq 2 with Utumi quotient ring U and C = Z(U) be the extended centroid of R. Suppose that 0 = p ∈ R, f (x1,. .. , xn) is a noncentral multilinear polynomial over C and F , G are two nonzero generalized derivations of R. If p(F(X)X − XG^2(X)) = 0 for all X\in f(R) then structure of maps are described.

In this article, we are intended to examine generalized skew-derivations that act as Jordan homoderivations on multilinear polynomials in prime rings. More specifically, we show that if F is generalized skew-derivation of a prime ring R with associated automorphism a such that the relation F(X^2) = F(X)^2 + F(X)X + XF(X) holds for all X 2 f (R), wh...

Let [Formula: see text] be a prime ring, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] the extended centroid of [Formula: see text], where [Formula: see text] is the Utumi quotient ring of [Formula: see text]. Suppose that [Formula: see text] are two nonzero generalized derivations of [Formula: see text]....

Let $R$ be a prime ring of characteristic different from 2, $n\geq 1$ a fixed integer, $C$ the extended centroid of $R$, $F$ a generalized skew derivation of $R$ and $L$ a Lie ideal of $R$. If there exists $0 \neq a \in R$ such that $a(F(xy)-yx)^{n}=0$ for all $x,y\in L$, then $L$ is central, unless $R$ satisfies the standard polynomial identity $s...

Let [Formula: see text] be a prime ring of characteristic different from [Formula: see text], [Formula: see text] its Utumi quotient ring, [Formula: see text] its extended centroid, [Formula: see text] a noncentral multilinear polynomial over [Formula: see text], [Formula: see text] and [Formula: see text] two generalized derivations of [Formula: s...

Let R be any non-commutative prime ring of char \((R)\ne 2\), L a non-central Lie ideal of R and F, G be b-generalized skew derivations of R. Suppose that for all \(u\in L\) and for some fixed integer \(n\ge 1\), then one of the following assertions holds: there exist \(a'',b''\in Q_r\) such that \(F(x)=xa''\), \(G(x)=b''x\) for all \(x\in R\) with...

Let [Formula: see text] be a prime ring of characteristic is not [Formula: see text], [Formula: see text] its right Martindale ring of quotient and [Formula: see text] be the extended centroid of [Formula: see text]. Suppose that [Formula: see text] is a noncentral multilinear polynomial over [Formula: see text] and [Formula: see text] with [Formul...

Let R denote a prime ring with a characteristic other than 2, $n\geq 1$ a fixed integer, $L$ a Lie ideal of $R$ and $C$ the extended centroid of $R$. Let $F$ and $G$ represent two generalized skew derivations of $R$ satisfying $(F(xy)-G(x)y-yx)^n=0$ for all $x,y\in L$, then $L$ is central, unless $R$ satisfies the standard polynomial identity $s_4(...

Let R be a prime ring of char \((R)\ne 2, 3\) and L a noncentral Lie ideal of R. Let U be the Utumi quotient ring of R and \(C=Z(U)\) be the extended centroid of R. Suppose that F, G, H are three generalized derivations of R such that for all \(u\in L\). Then either R satisfies standard polynomial \(s_4(x_1,x_2,x_3,x_4)\) or one of the following ho...

Let R be a prime ring with char(R) 6= 2. Suppose that f(x1,... , xn) be a noncentral multilinear polynomial over C, G be nonzero generalized derivation of R and d a nonzero derivation of R. In this paper we describe all possible forms of G in the given case.

Let R be a semiprime ring with center Z(R), \(\lambda \) be a nonzero left-sided ideal of R, \(0 \ne a\in R\) and \(F, G: R\rightarrow R\) be multiplicative (generalized)-derivations of R associated to the maps \(d,g:R\rightarrow R\), respectively. In the present paper, we study the following identities: \(a(G(xy)\pm F(x)F(y)\pm xy) \in Z(R)\); \(a...

Let R be a prime ring with extended centroid C and characteristic of R be different from 2. Suppose that U is the Utumi quotient ring of R and \({\mathcal {L}} \) a noncentral Lie ideal of R. Assuming \({\mathcal {T}} _1, {\mathcal {T}} _2, {\mathcal {T}} _3\) three generalized derivations of R, we investigate the identity for all \(u\in {\mathcal...

Let R be a prime ring, let 0 ≠ b ∈ R {0\neq b\in R} , and let α and β be two automorphisms of R . Suppose that F : R → R {F:R\rightarrow R} , F 1 : R → R {F_{1}:R\rightarrow R} are two b -generalized ( α , β ) {(\alpha,\beta)} -derivations of R associated with the same ( α , β ) {(\alpha,\beta)} -derivation d : R → R d:R\rightarrow R , and let G :...

This paper represents the structures of generalized derivations satisfying an identity with [Formula: see text]-Engel condition. To prove our result, we take [Formula: see text] as a prime ring with [Formula: see text], and [Formula: see text] as Utumi quotient ring of [Formula: see text]. The center of [Formula: see text] is [Formula: see text] wh...

UDC 512.5 Suppose that R is a prime ring with c h a r ( R ) ≠ 2 and f ( ξ 1 , … , ξ n ) is a noncentral multilinear polynomial over C ( = Z ( U ) ) , where U is the Utumi quotient ring of R . An additive mapping h : R → R is called homoderivation if h ( a b ) = h ( a ) h ( b ) + h ( a ) b + a h ( b ) for all a , b ∈ R . We investigate the behavior...

Suppose that R is a prime ring with Utumi quotient ring U, extended centroid C and \(f(\xi _1,\ldots ,\xi _n)\) a noncentral multilinear polynomial over C. Let I be a nonzero ideal of R and char \((R)\ne 2\). If F, G and H are three generalized derivations of R such that $$\begin{aligned} F\Big ([G(f(\xi )), f(\xi )]\Big )=H(f(\xi )^2) \end{aligned...

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...

Let $\mathscr{R}$ be a noncommutative prime ring of characteristic different from $2$, $\mathscr{Q}_{r}$ the right Martindale quotient ring of $\mathscr{R}$ and $\mathscr{C}=\mathscr{Z}(\mathscr{Q}_r)$ the extended centroid of $\mathscr{R}$. The paper studies the relationship between the structure of prime rings and the behaviour of generalized- sk...

Let R be a prime ring of \(char(R)\ne 2\), U and C the Utumi ring of quotients and the extended centroid of R, respectively, F and H two generalized derivations of R, d non-zero derivation of R and \(f(\xi _1,\ldots , \xi _n)\) a non-central multilinear polynomial over C. Denote by f(I) the set \(\{f(x_1,\ldots ,x_n):x_1,\ldots ,x_n\in I\}\) of all...

Let [Formula: see text] be a prime or semiprime ring, [Formula: see text] a nonzero left-sided ideal of [Formula: see text] and [Formula: see text] be two automorphisms of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two generalized [Formula: see text]-derivations of [Formula: see text] associated with [Formula: see text]...

Let R be a prime ring of characteristic different from 2, Z(R) its center, Qr its right Martindale quotient ring, C its extended centroid, f(x1,…,xn) a non-central multilinear polynomial over C, b∈Qr and F, G two b-generalized skew derivations of R. If there exists a∈R\Z(R) such that [a,F(f(r1,…,rn))f(r1,…,rn)]=f(r1,…,rn)G(f(r1,…,rn)) for any r1,…,...

Let R be a prime ring of characteristic different from 2, U its Utumi quotient ring, C its extended centroid, L a non-central Lie ideal of R and F, G and H three generalized derivations of R. If $$\begin{aligned}{}[F(u)G(u)-uH(u),u]=0 \end{aligned}$$for all \(u \in L\), then one of the following holds: (1) there exist \(a,c,q,p,p'\in U\) such that...

Let R be a noncommutative prime ring of characteristic different from 2, Qr be the right Martindale quotient ring of R and C=Z(Qr) be the extended centroid of R. Suppose that f(x1,…,xn) is a noncentral multilinear polynomial over C and 0≠F, G are two generalized skew-derivations of R associated to the same automorphism α. If F(u2)=F(u)G(u) for all...

Let R be a semiprime ring with center Z(R) and \(\lambda \) a nonzero left ideal of R. A mapping \(F: R\rightarrow R\) (not necessarily additive) is said to be a multiplicative (generalized)-derivation on R, if there exists a map d (not necessarily an additive map or derivation) on R such that \(F(xy)=F(x)y+xd(y)\) holds for all \(x,y\in R\). Suppo...

Let R be a prime ring of char \((R)\ne 2\), \(Q_r\) its right Martindale quotient ring and C its extended centroid, \(f(x_1,\dots , x_n)\) a multilinear polynomial over C that is noncentral-valued on R and F an X-generalized skew derivation of R. If for some \(0\ne a\in R\), $$\begin{aligned} a[F(f(x_1,\dots , x_n)),f(x_1,\dots , x_n)]\in C \end{al...

Let R be a prime ring of char(R)≠ 2, U its Utumi ring of quotients and center C = Z(U) its extended centroid, I a both sided ideal of R, f(x1,…,xn) a multilinear polynomial over C, that is noncentral-valued on R, F, G be two generalized derivations of R and d be a derivation of R. Let f(I) be the set of all evaluations of the multilinear polynomial...

Let $R$ be a prime ring of char $(R)\neq 2$, $U$ its Utumi ring of quotients and center $C=Z(U)$ its extended centroid, $I$ a both sided ideal of $R$, $f(x_{1},\ldots,x_{n})$ a multilinear polynomial over $C$, that is, noncentral-valued on $R$, $F$, $G$ be two generalized derivations of $R$ and $d$ be a derivation of $R$. Let $f(I)$ be the set of a...

Let R be a prime ring of characteristic different from 2, \(Q_r\) be the right Martindale quotient ring of R and \(C=Z(Q_r)\) be the extended centroid of R. Suppose that \(f(x_1,\ldots ,x_n)\) is a noncentral multilinear polynomial over C and F, G are two nonzero generalized skew-derivations of R associated to the same automorphism of R. If $$\begi...

Let R be a noncommutative prime ring of characteristic different from 2 and 3, C the extended centroid of R, F and G two generalized derivations of R, d a nonzero derivation of R and \(f(x_1,\ldots ,x_n)\) a multilinear polynomial over C. Suppose that I is a nonzero ideal of R and \(f(I)=\{f(x_1,\ldots ,x_n)| x_1,\ldots ,x_n\in I\}\). If \(f(x_1,\l...

Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy) + G(yx)) n = (xy ± yx) for all x, y ∈ I, then one of the following holds: 1. R is commutative; 2. n = 1 and H(x) = x and G(x) = ±x for all x ∈ R. Moreover, we examine the case where R is...

Let R be a noncommutative prime ring of char (R)? 2 with Utumi quotient ring U and extended centroid C and I a nonzero two sided ideal of R. Suppose that F(? 0), G and H are three generalized derivations of R and f (x1,...,xn) is a multilinear polynomial over C, which is not central valued on R. If F(G(f(r))f(r)- f(r)H(f(r))) = 0 for all r = (r1,.....

Let R be a semiprime ring and \(\lambda\) be a nonzero left ideal of R. A mapping \(d : R \rightarrow R\) (not necessarily a derivation nor an additive map) which satisfies \(d(xy) = d(x)y+xd(y)\) for all \(x, y \in R\) is called a multiplicative derivation of R. A mapping \(F:R\rightarrow R\) (not necessarily additive) is called a multiplicative (...

In this paper, we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan centralizers raised in Ali and Fošner (Algebra Colloq 21:411–420, 2014) and Fošner (Demonstr Math 46:254–262, 2013). Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that ev...

Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$ and $f(r_1,\ldots,r_n)$ be a multilinear polynomial over $C$, which is not central valued on $R$. Suppose that $F$ and $G$ are two nonzero generalized derivations of $R$ such that $G\neq Id$, identity map and $$F(f(r)^2...

Let R be a prime ring with characteristic different from 2 and 3, Qr be its right Martindale quotient ring, C be its extended centroid, F a nonzero generalized skew derivation of R and L a noncentral Lie ideal of R. If [[F(u),u],F(u)]∈Z(R)for all u∈L, then one of the following holds:there exists λ∈C such that F(x)=λx for all x∈R;R satisfies s4 and...

Let R be a noncommutative prime ring, I a nonzero left ideal of R, L a non-central Lie ideal of R, U the left Utumi quotient ring of R and C=Z(U) the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and m,n,r,k fixed positive integers. In the present paper, we describe the structure of R and all possible for...

Let R be a ring. An additive mapping \(F : R\rightarrow R\) is called a generalized derivation if there exists a derivation \(d : R\rightarrow R \) such that \( F(x y) = F(x)y + xd(y)\) for all \( x, y \in R\). In this paper, first we describe the structure of prime rings involving automorphisms and then characterized generalized derivations on sem...

Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U, extended centroid C, f(x 1 , … , x n ) 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 , … , x n ) in R. In the present paper, we study b[...

Let R be a prime ring of characteristic \(\ne 2\), \(Q_r\) its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) a generalized skew derivation of R, \(f(x_1,\ldots ,x_n)\) a multilinear polynomial over C not central-valued on R and S the set of all evaluations of \(f(x_1,\ldots ,x_n)\) in R. If \(a[F(x),x]\in C\) for all \(x\in S\...

Let $n\geq 1$ be a fixed integer, $R$ a prime ring with its right Martindale quotient ring $Q$, $C$ the extended centroid, and $L$ a Lie ideal of $R$. Suppose that $\alpha$ is an automorphism of $R$ and $F\neq0$ a generalized skew derivation of $R$ with the associated automorphism $\alpha$. In the paper, we investigate the situation $(F(x)F(y)-yx)^...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid, f (X1, . . . , Xn) a multilinear polynomial over C that is noncentral-valued on R and F a generalized skew derivation of R. If for some 0 ̸= a ∈ R − C, a[F (f (r)), f (r)] − [F (f (r)), f (r)]a ∈ Z(R) for all r = (r1, . . . ,...

Let R be a noncommutative prime ring of characteristic different from 2, Qr the right Martindale quotient ring of R, C=Z(Qr) the extended centroid of R and f(x1,…,xn) a noncentral multilinear polynomial over C. In this paper we describe all possible forms of two generalized skew derivations F and G of R satisfying the condition F(u)2=G(u2), for all...

Let R be a noncommutative prime ring of characteristic dif- ferent from 2, U the Utumi quotient ring of R, C the extended centroid of R and f(x 1 , . . ., 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 , . . ., x n ) on R. If d is a nonzero derivation of R and G a...

In this paper we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan cen-tralizers raised in [S. Ali and A. Fošner, On Generalized (m, n)-Derivations and Generalized (m, n)-Jordan Derivations in Rings, Algebra Colloq. 21 (2014), 411-420] and [A. Fošner, A note on generalized (m, n)-Jo...

In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fo\v{s}ner,...

Let R be a prime ring with characteristic different from 2, Qr be its right Martindale quotient ring, C be its extended centroid, F a nonzero generalized skew derivation of R, L a noncentral Lie ideal of R, n ≥ 1, m ≥ 0 be fixed integers and 0 ≠ a ∈ R. If aumF(u)ⁿ = 0 for all u ∈ L, then one of the following holds: (i) m = 0 and there exists b ∈ Qr...

Let $R$ be a 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 $C$, $0\neq p\in R$, $F$ and $G$ be two $b$-generalized derivations of $R$. In this paper we describe all possible forms of $F$ and $G...

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 $f(x_1, \ldots , x_n)$ a non-central multilinear polynomial over $C$. Let $F$ and $G$ be two generalized derivations of $R$ and $d$ a nonzero derivation of $R$ such that $$d([F(f(r_1, \ldots , r_n)), f(r_1, \...

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(x1, …, xn) be a noncentral multilinear polynomial over C, b ∈ 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 = (r1, …, r...

Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$, $F$, $G$ and $H$ three generalized derivations of $R$, $I$ an ideal of $R$ and $f(x_1,\ldots,x_n)$ be a multilinear polynomial over $C$ which is not central valued on $R$. If $$F(f(r))G(f(r))=H(f(r)^2)$$ for all $r=(r_...

Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) be the extended centroid of R, H and G two generalized derivations of R, L a noncentral Lie ideal of R, I a nonzero ideal of R. The left annihilator of S ⊆ R is denoted by lR(S) and defined by lR(S) = {x ∈ R| xS = 0}. Suppose that S = {H(un)un +unG(un) | u ∈ L} and T = {H(xn)xn +xnG...

Let $R$ be a prime ring with its Utumi ring of quotients $U$, $C = Z(U)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$ and $0\neq a\in R$. If $R$ admits a generalized derivation $F$ such that $a(F(u^2)\pm F(u)^{2})=0$ for all $u\in L$, then one of the following holds: \begin{enumerate} \item there exists $b\in U$ such that $F(x)=...

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R and F a generalized derivation with associated derivation d of R. Suppose that (F([x, y])) m = [x, y] n for all x, y ∈ I, a nonzero ideal of R, where m ≥ 1 and n ≥ 1 are fixed integers, then one of the following holds: (1) R is commutative; (2) there exists a...

Let $R$ be a noncommutative prime ring with its Utumi ring of quotients $U$, $C=Z(U)$ the extended centroid of $R$, $G$ a generalized derivation of $R$ and $I$ a nonzero ideal of $R$. If $I$ satisfies any one of the following conditions: (i) $$\bigg[\bigg[G([x,y]_n), [x,y]_n\bigg], G([x,y]_n)\bigg]\in C,$$ (ii) $$\bigg(G(x \circ_n y) \circ (x \ci...

Let R be a semiprime ring. An additive mapping F : R → R is called a generalized derivation of R if there exists a derivation d : R → R such that F ( xy ) = F ( x ) y + xd ( y ) holds, for all x , y ∈ R . The objective of the present paper is to study the following situations: (1) [ d ( x ), F ( y )] = ±[ x , y ]; (2) [ d ( x ), F ( y )] = ± x ο y...

For a ring $R$ with an automorphism $\alpha$ an $n$-additive mapping $D:R^n\longrightarrow R$ is called a skew $n$-derivation w.r.t. $\alpha$ if it is an $\alpha$-derivation of $R$ for each argument. Namely it is always an $\alpha$-derivation of $R$ for the argument being left once $(n-1)$ arguments are fixed by $(n-1)$ elements in $R$. In the pres...

\begin{abstract} Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$, and $f(x_1,\ldots,x_n)$ be a multilinear polynomial over $C$, which is not central valued on $R$. Suppose that $F$ and $G$ are two nonzero generalized derivations of $R$ such that $G([F(f(r_1,\ldots,r_...

Let R be a prime ring, I a nonzero right ideal of R, U the two sided Utumi quotient ring of R, C

Let R be a prime ring and \(F, G: R\rightarrow R\) be two generalized derivations of R such that \(F^2+G\) is n-commuting or n-skew-commuting on a nonzero square closed Lie ideal U of R. In the present paper we prove under certain conditions that \(U\subseteq Z(R)\).

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 $R$. If $d$ is a nonzero derivation of $R$ and $F$ is a generalized derivation of $R$ such that $$d\Big(\Big[F^2(f(x_1, \ldots, x_n)), f(x_1, \l...

Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) extended centroid of R, F a nonzero generalized derivation of R, L a noncentral Lie ideal of R and k ≥ 2 a fixed integer. Suppose that there exists 0 ≠ a ∈ R such that a[F(un1), un2, ⋯, unk] = 0 for all u ∈ L, where n1, n2, ⋯, nk ≥ 1 are fixed integers. Then either there exists λ ∈...

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 \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C which is not central valued on R. If $$\begin{aligned} {[}F(u)u, G(v)v]=0 \end{aligned}$$for all \(u,v\in f(R)\)...

Let $R$ be a ring with center $Z(R)$. A mapping $F:R\rightarrow R$ is called a multiplicative generalized derivation, if $F(xy)=F(x)y+xg(y)$ is fulfilled for all $x,y\in R$, where $g:R\rightarrow R$ is a derivation. In the present paper, our main object is to study the situations: (1) $F(xy)- F(x)F(y)\in Z(R)$, (2) $F(xy)+ F(x)F(y)\in Z(R)$, (3) $F...

Let $R$ be a semiprime ring and $F: R\rightarrow R$ a mapping such that $F(xy)=F(y)x+yd(x)$ for all $x, y\in R$, 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

Let R be a prime ring of characteristic not equal to 2, U the Utumi quotient ring of R, \(C=Z(U)\) the extended centroid of R, and \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that \(F(f(x_1,\ldots ,x_n))G(f(x_1,\ldots ,x_n))+G(f(x_1,\ldots...

Let R be a prime ring with center Z(R) and with extended centroid C, d a derivation of R and f (x1, . . . , xn) a nonzero multilinear polynomial over C, m ≥ 1 and p ≥ 1 two integers. In the present paper, we study the situations (i) ((d(f (x1, . . . , xn))) m − f (x1, . . . , xn)) p = 0; (ii) ((d(f (x1, . . . , xn))) m − f (x1, . . . , xn)) p ∈ Z(R...

Let R be a prime ring with center Z(R), a R (a 0) and I a nonzero ideal of R. Suppose that F,d:R→R${F,d\colon R\rightarrow R}$ are any two mappings such that F(xy)=F(x)y+xd(y)${F(xy) = F(x)y+xd(y)}$ for all x,yR${x, y \in R}$ . For all x,yI${x,y\in I}$ , we investigate the identities a(F(xy)±xy)=0${a(F(xy)\pm xy)=0}$ , a(F(xy)±yx)=0${a(F(xy)\pm yx)...

Let R be a prime ring with Utumi quotient ring U and extended centroid C, F a nonzero generalized derivation of R, I a nonzero right ideal of R, \(f(r_1,\ldots ,r_n)\) a multilinear polynomial over C and \(s\ge 1, t\ge 1\) be fixed integers. If \((F(f(r_1,\ldots ,r_n))^s-f(r_1,\ldots ,r_n)^s)^t=0\) for all \(r_1,\ldots ,r_n\in I\), then one of the...

Let R be a semiprime ring and � any mapping on R. A mapping F : R --->R is called multiplicative (generalized)-derivation if F(xy) = F(x)y+xd(y) for all x; y \in 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.

Let $R$ be a prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$, $F$ and $G$ two nonzero generalized derivations of $R$, $I$ an ideal of $R$ and $f(x_1,\ldots,x_n)$ a multilinear polynomial over $C$ which is not central valued on $R$. If $$F(G(f(x_1,\ldots,x_n))f(x_1,\ldots,x_n))=0$$ for all $x_1,...

Let $n$ be a fixed positive integer, $R$ be a prime ring, $D$ and $G$ two derivations of $R$, $L$ a noncentral Lie ideal of $R$. Suppose that there exists $0\neq a\in R$ such that $a(D(u)u^n-u^nG(u))=0$ for all $u \in L$, where $n\geq 1$ is a fixed integer. Then one of the following holds: \begin{enumerate} \item $D=G=0$, unless $R$ satisfies $s...

Let $R$ be a prime ring with center $Z(R)$ and extended centroid $C$, $H$ a non-zero generalized derivation of $R$ and $n\geq 1$ a fixed integer. In this paper we study the situations: (1) $H(u^2)^n-H(u)^{2n}\in C$ for all $u \in L$, where $L$ is a non-central Lie ideal of $R$; (2) $H(u^2)^n-H(u)^{2n}=0$ for all $u\in [I, I]$, where $I$ is a nonzer...

Let R be a prime ring with center Z(R), I a non-zero ideal of R and α:R→R any mapping on R. Suppose that G and F are two generalized derivations associated with derivations g and d respectively on R. In this paper we study the following situations: (i) G(xy)±F(x)F(y)±xy∈Z(R), (ii) G(xy)±F(y)F(x)±xy∈Z(R), (iii) G(xy)±F(x)F(y)±yx∈Z(R), (iv) G(xy)±F(y...

Let R be a noncommutative prime ring with its Utumi ring of quotients U, C = Z(U) the extended centroid of R, F a generalized derivation of R and I a nonzero ideal of R. Suppose that there exists 0 not equal a is an element of R such that a(F([x, y])(n) - [x, y]) = 0 for all x, y is an element of I, where n >= 2 is a fixed integer. Then one of the...

Let R be a prime ring with its Utumi ring of quotients U and extended centroid C. Suppose that F is a generalized derivation of R and L is a noncentral Lie ideal of R such that F(u)[F(u), u]n = 0 for all u ∈ L, where n ⩾ 1 is a fixed integer. Then one of the following holds: (1) there exists λ ∈ C such that F(x) = λx for all x ∈ R (2) R satisfies s...

Let R be a ring. A map F : R ! R is called a multiplicative (generalized)-derivation if F(xy) = F(x)y + xg(y) is fulfilled for all x, y 2 R where g : R ! R is any map (not necessarily derivation). Let S be a nonempty subset of R. In the present paper for various choices of S we study the following situations: (i) F([x, y]) = ±(xy ± yx), (ii) F(x �...

Let R be a semiprime ring with center Z(R). A mapping F: R → R (not necessarily additive) is said to be a multiplicative (generalized)-derivation if there exists a map f: R → R (not necessarily a derivation nor an additive map) such that F(xy) = F(x)y + xf(y) holds for all x; y ϵ R. The objective of the present paper is to study the following ident...