Search

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]-derivations [Formula: see text] and [Formula: see text], respectively. The purpose of this paper is to investigate the following identities: (1) [Formula: see text]; (2) [Formula: see text]; (3) [Formula: see text]; (4) [Formula: see text]; (5) [Formula: see text]; (6) [Formula: see text]; (7) [Formula: see text]; (8) [Formula: see text]; (9) [Formula: see text]; (10) [Formula: see text]; (11) [Formula: see text]; (12) [Formula: see text], for all [Formula: see text].

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 $$\begin{aligned} F(u)u-G(u)F(u)=0 \end{aligned}$$ for all $$u\in f(R)$$ , then one of the following holds

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 $$f(x_1,\ldots ,x_n)$$ is not central valued on R and $$\begin{aligned} d(F^2(u)u-G(u^2))=0 \end{aligned}$$ for all $$u\in f(I)$$ , then all possible forms of the maps F, G and d are described in the present paper.

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 ofthe 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 a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.

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,..., rn) ? In, then we obtain information about the structure of R and describe the all possible forms of the maps F, G and H. This result generalizes many known results recently proved by several authors ([1], [4], [5], [8], [9], [13], [15]).

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 (generalized)-derivation of R if there exists a map $$d:R \rightarrow R$$ (not necessarily a derivation nor an additive map) such that $$F(xy)=F(x)y+xd(y)$$ holds for all $$x,y\in R$$ . The objective of this paper is to study the following identities: (1) $$[d(x),F(y)]=\pm [x,y]$$ , (2) $$[d(x),F(y)]=\pm x\circ y$$ , (3) $$[d(x),F(y)]=0$$ , (4) $$F([x,y])\pm [\delta (x),\delta (y)]\pm [x,y]=0$$ , (5) $$d'([x,y])\pm [\delta (x),\delta (y)]\pm [x,y]=0$$ , (6) $$d'([x,y])\pm [\delta (x),\delta (y)]=0$$ , (7) $$F(x \circ y)\pm \delta (x)\circ \delta (y) \pm x\circ y=0$$ , (8) $$d'(x \circ y)\pm \delta (x)\circ \delta (y) \pm x\circ y=0$$ , (9) $$d'(x \circ y)\pm \delta (x)\circ \delta (y)=0$$ , for all $$x,y\in \lambda$$ , where F is a multiplicative (generalized)-derivation of R associated to the map d, and $$d'$$ , $$\delta$$ are multiplicative derivations of R.

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 Fosner (Algebra Colloq 21:411–420, 2014) and Fosner (Demonstr Math 46:254–262, 2013). Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized (m, n)-Jordan derivation (resp., a generalized (m, n)-Jordan centralizer) is a derivation (resp., a two-sided centralizer).

Let R be a prime ring with characteristic different from 2 and 3, $$Q_r$$ 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 $$\begin{aligned}{}[[F(u),u],F(u)]\in Z(R) \end{aligned}$$for all $$u\in L$$, then one of the following holds

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 ...

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 semiprime rings which satisfy certain differential identities. As applications, and apart from proving the other results, many known theorems can be either generalized or deduced. Moreover, we apply our results to functional analysis, and to study the analogous conditions for continuous linear generalized derivations on Banach algebras.

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-va...

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 noncentral multilinear polynomial over $C$. If $F$, $G$ and $H$ are three nonzero generalized derivations of $R$ such that \[ F\big( G(f(X)) f(X) \big) = f(X) H(f(X)) \] for all $X = (x_{1},\ldots,x_{n}) \in R^n$, then we describe the nature of the maps $F$, $G$ and $H$.

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 th...

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. Further, assume that F and G are two generalized derivations of R and d is a nonzero derivation of R. De Filippis (Isr J Math 162:93–108, 2007) studied the situation $$[F(f(r_1, \ldots , r_n)),f(r_1, \ldots , r_n)]=0$$ for all $$r_1, \ldots , r_n\in R$$ , and found the structure of F and R. Recently, De Filippis et al. (Commun Algebra 40:1918–1932, 2012) investigated the situation $$d([F(f(r_1, \ldots , r_n)), f(r_1, \ldots , r_n )])=0$$ for all $$r_1,\ldots , r_n\in R$$ , and obtained the nature of mappings and R. In this paper, we generalize both the results given in (2007, 2012) by considering the situation $$d([F(f(r_1, \ldots , r_n)), f(r_1, \ldots , r_n)]) = [G(f(r_1, \ldots , r_n)), f(r_1, \ldots , r_n)]$$ for all $$r_1, \ldots , r_n\in R$$ .

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x1,..., x n ) 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 = (r1,..., r n ) ∈ I n , then one of the following conditions holds

Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U, extended centroid C, $$f(x_1,\ldots ,x_n)$$ 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 $$f(x_1,\ldots ,x_n)$$ in R. In the present paper, we study $$b[d(u),u]+p[d(u),u]q+[d(u),u]c=0$$ for all $$u\in f(R)$$ , which includes left-sided, right-sided as well as two-sided annihilating conditions of the set $$\{[d(u),u] : u\in f(R)\}$$ . 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 $$\begin{aligned} F^2([d(u), u])=0 \end{aligned}$$ for all $$u\in f(R)$$ , then there exists $$a\in U$$ with $$a^2=0$$ such that $$F(x)=xa$$ for all $$x\in R$$ or $$F(x)=ax$$ for all $$x\in R$$ .

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 $$x\in S$$ , then there exist $$\lambda \in C$$ and $$b\in Q_r$$ such that $$F(x)=bx+xb+\lambda x$$ , for all $$x\in R$$ and one of the following holds

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 polyn...

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, f(x1,…,xn) 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([F2(f(x1,…,xn)),f(x1,…,xn)])=0 for all x1,…,xn∈R, 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.

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 \(u,v\in f(R)\), then there exist \(a,b\in U\) such that \(F(x)=ax\) and \(G(x)=bx\) for all \(x\in R\) with \([a, b]=0\) and \(f(x_1,\ldots ,x_n)^2\) is central valued on R.

Let R be a prime ring with characteristic different from 2, let U be its right Utumi quotient ring, let C be its extended centroid, let F and G be additive maps on R, letf(x 1 , . . . ,x n ) be a multilinear polynomial over C, and let I be a nonzero right ideal of R. We obtain information about the structure of R and describe the form of F and G in the following cases

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(x1,…, xn) a multilinear polynomial over C which is not central valued on R. If 0 ≠ a ∈ R such that for all u, v ∈ f(I), where f(I) is the set of all evaluations of f(x1,…, xn) in I, then there exists b ∈ U such that F(x) = bx for all x ∈ R and one of the following statements holds: f(x1,…, xn)2 is central valued on R;ab = 0.

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 ( x y ) = F ( x ) y + x d ( y ) ${F(xy) = F(x)y+xd(y)}$ for all x , y ∈ R ${x, y \in R}$ . For all x , y ∈ I ${x,y\in I}$ , we investigate the identities a ( F ( x y ) ± x y ) = 0 ${a(F(xy)\pm xy)=0}$ , a ( F ( x y ) ± y x ) = 0 ${a(F(xy)\pm yx)=0}$ , a ( F ( x ) F ( y ) ± x y ) = 0 ${a(F(x)F(y)\pm xy)=0}$ , a ( F ( x ) F ( y ) ± y x ) = 0 ${a(F(x)F(y)\pm yx)=0}$ , a ( d ( x ) F ( y ) ± x y ) ∈ Z ( R ) ${a(d(x)F(y)\pm xy)\in Z(R)}$ , a ( d ( x ) F ( y ) ± y x ) ∈ Z ( R ) ${a(d(x)F(y)\pm yx)\in Z(R)}$ and a ( F ( x y ) ± F ( x ) F ( y ) ) = 0 ${a(F(xy)\pm F(x)F(y))=0}$ .

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x1,…, xn) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) − f(r)G(f(r))) = 0 for all r = (r1,…, rn) ∈ Rn, then one of the following holds: There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a − q] = 0 and f(x1,…, xn)2 is central valued on R;There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa − bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;R satisfies s4 and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa − bx and G(x) = ax + xb for all x ∈ R;There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb − δ(x), G(x) = bx + δ(x) and d(x) = [c, x...

Let R be a prime ring with its Utumi ring of quotients U, F a nonzero generalized derivation of R and L a noncentral Lie ideal of R. Suppose that $$[F(u^{n_1}),u^{n_2},u^{n_3},\ldots ,u^{n_k}]=0$$ for all $$u \in L$$ , where $$n_1, n_2, \ldots ,n_k\ge 1$$ are fixed integers. Then one of the following holds: Also we study the situation, when $$x\in [I,I]$$ , where I is a nonzero left ideal of R. As an application we obtain some range inclusion results of continuous generalized derivations on Banach algebras.

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 \(F(f(x_1,\ldots ,x_n))G(f(x_1,\ldots ,x_n))+G(f(x_1,\ldots ,x_n))F(f(x_1,\ldots ,x_n))=0\) for all \(x_1,\ldots ,x_n\in R\). Then \(f(x_1,\ldots ,x_n)^2\) is central valued on R and one of the following holds: (1) there exist \(0\ne \lambda \in C\) and \(p\in U-C\), such that \(F(x)=\lambda x\) for all \(x\in R\) and \(G(x)=[p,x]\) for all \(x\in R\); (2) there exist nonzero \(a, b \in U\), such that \(F(x)=xa\) for all \(x\in R\) and \(G(x)=bx\) for all \(x\in R\) with \(ab=-ba\in C\); (3) there exist nonzero \(a, b \in U\), such that \(F(x)=ax\) for all \(x\in R\) and \(G(x)=xb\) for all \(x \in R\) with \(ab=-ba\in C\); (4) there exist \(0\ne \lambda \in C\) and \(a\in U-C\), such that \(F(x)=[a,x]\) for all \(x\in R\) and \(G(x)=\lambda x\) for all \(x\in R\).

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 [Formula: see text] is a non-central multilinear polynomial over C, [Formula: see text], and F, G are two b-generalized derivations of R. In this paper we describe all possible forms of F and G in the case [Formula: see text] for all [Formula: see text] in Rn.

Let R be a semiprime ring and \(\alpha \) any mapping on R. A mapping \(F:R\rightarrow R\) is called multiplicative (generalized)-derivation if \(F(xy)=F(x)y+xd(y)\) for all \(x, y \in R\), where \(d:R\rightarrow R\) 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, the 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 $$\begin{aligned} F(G(f(x_1,\ldots ,x_n))f(x_1,\ldots ,x_n))=0 \end{aligned}$$ for all \(x_1,\ldots ,x_n \in I\), then one of the followings holds: (1) there exist \(a,b\in U\) such that \(F(x)=ax\) and \(G(x)=bx\) for all \(x\in R\) with \(ab=0\); (2) there exist \(a,b,p\in U\) such that \(F(x)=ax+xb\) and \(G(x)=px\) for all \(x\in R\) with \(F(p)=0\) and \(f(x_1,\ldots ,x_n)^2\) is central valued on R. We also obtain some related results in cases where R is a semiprime ring and Banach algebra.

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

Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F,G be two generalized derivations of R with associated derivations d and δ of R respectively. In the present paper, we study the situations (1) F(u)G(v)-uv ∈ Z(R), (2) F(u)G(v)+uv ∈ Z(R), (3) d(u)F(v)-vu ∈ Z(R), (4) d(u)F(v) + vu ∈ Z(R); for all u, v ∈ U and show that if d≠0 and δ≠0, then U ⊆ Z(R).

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 $$F(f(r)^2)=F(f(r))G(f(r))+G(f(r))F(f(r))$$ for all $r=(r_1,\ldots,r_n)\in R^n$. Then one of the following holds:(1) there exist $\lambda \in C$ and $\mu \in C$ such that $F(x)=\lambda x$ and $G(x)=\mu x$ for all $x\in R$ with $2\mu=1$;(2) there exist $\lambda \in C$ and $p,q\in U$ such that $F(x)=\lambda x$ and $G(x)=px+xq$ for all $x\in R$ with $p+q\in C$, $2(p+q)=1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(3) there exist $\lambda \in C$ and $a\in U$ such that $F(x)=[a,x]$ and $G(x)=\lambda x$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(4) there exist $\lambda \in C$ and $a,b\in U$ such that $F(x)=ax+xb$ and $G(x)=\lambda x$ for all $x\in R$ with $a+b\in C$, $2\lambda =1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(5) there exist $a, p\in U$ such that $F(x)=xa$ and $G(x)=px$ for all $x\in R$, with $(p-1)a=-ap\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;(6) there exist $a, q\in U$ such that $F(x)=ax$ and $G(x)=xq$ for all $x\in R$ with $a(q-1)=-qa\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$.

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(x1, . . . , xn) be a multilinear polynomial over C which is not central valued on R. If F (f(x1, . . . , xn))f(x1, . . . , xn)− f(x1, . . . , xn)G(f(x1, . . . , xn)) = 0 for all x1, . . . , xn ∈ I, then one of the following holds: (1) F (x) = xa and G(x) = xb for all x ∈ R with a = b ∈ C; (2) F (x) = xa and G(x) = bx for all x ∈ R with a = b; (3) F (x) = ax and G(x) = xb for all x ∈ R with a = b ∈ C; (4) F (x) = ax and G(x) = xb for all x ∈ R with a = b and f(x1, . . . , xn) is central valued on R; (5) F (x) = ax and G(x) = bx for all x ∈ R, with a = b ∈ C. We finally extend the result to a semiprime ring R in case F (x)x − xG(x) = 0 for all x ∈ R. Throughout this paperR always denotes an associative prime ring with center Z(R), extended centroid C and U its Utumi quotient ring. The Lie commutator of x and y is denoted by [x, y] and defined by [x, y] = xy − yx for x, y ∈ R. An Mathematics Subject Classification: 16N60, 16W25 .

Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R, fpx1, . . . , xnq a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that Gpfprqqfprq fprqHpfprqq P C, for all r “ pr1, . . . , rnq P I. Then one of the following holds: 1. there exist a, b, p P U and α P C such that Gpxq “ ax ` rp, xs and Hpxq “ bx, for all x P R, and pa bqI “ p0q “ pa` p αqI; 2. R satisfies s4, the standard identity of degree 4, and there exist a, a1 P U , α, β P C such that Gpxq “ ax` xa1 ` αx and Hpxq “ a1x xa` βx, for all x P R; 3. R satisfies s4 and there exist a, a1 P U , and d : R N R, a derivation of R, such that Gpxq “ ax` dpxq and Hpxq “ xa1 dpxq, for all x P R, with a` a1 P C; 4. R satisfies s4 and there exist a, a1 P U , and d : R N R, a derivation of R, such that Gpxq “ xa` dpxq and Hpxq “ ax1 dpxq, for all x P R, with a a1 P C; 5. there exists e “ e P SocpRCq such that I “ eR and one of the following holds: (a) rfpx1, . . . , xnq, xn`1sxn`2 is an identity for I; (b) charpRq “ 2 and s4px1, x2, x3, x4qx5 is an identity for I; (c) rfpx1, . . . , xnq, xn`1sxn`2 is an identity for I and there exist a, a1, b, b1 P U , α P C and d : R N R, a derivation of R, such that Gpxq “ ax ` xa1 ` dpxq, Hpxq “ bx` xb1 dpxq, for all x P R, with pa b1 αqI “ p0q “ pb a1 αqI.

Let R be an associative ring. A mapping f : R −→ R is said to be additive if f (x +y) = f (x) +f (y) holds for all x;y ∈ R: An additive mapping d : R −→ R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x;y ∈ R: In this paper, we investigate commutativity of prime and semiprime rings satisfying certain identities involving additive mappings and derivations. Moreover, some results have also been discussed.