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Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

Let R be a prime ring of characteristic different from two, Q be the Martindale quotient ring of R and C be the extended centroid of R. Suppose that d is a derivation of R, I a nonzero ideal of R and \(m,k\in \mathbb {Z^{+}}\) where \(m > 1\). If $$[(x\circ _k y)^d,x\circ _k y]^m=[(x\circ _k y)^d,x\circ _k y]$$ for all \(x, y \in I\), then either \(d = 0\) or R satisfies \(S_{4}\), the standard identity in four variables. We also extend the result to semiprime rings.

In this manuscript, we discuss the behaviour and nature of generalized derivation $${\mathscr {G}}$$ on a (semi-) prime ring $${\mathscr {R}}$$ satisfying certain differential identities over $${\mathscr {I}}$$ , a nonzero ideal of $${\mathscr {R}}$$ . Moreover, we extend our purely ring theoretic result to a non-commutative Banach algebras and obtained some range inclusion results of continuous generalized derivations.

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Let X be a real or complex Banach space, let L ( X ) be the algebra of all bounded linear operators of X and let A ( X ) ⊆ L ( X ) be a standard operator algebra. Suppose there exists a linear mapping T : A ( X ) → L ( X ) satisfying the relation T(An) = T(A)An−1 − AT(An−2)A − An−1T(A) for all A ∈ A ( X ) , where n > 2 is some fixed integer. Then T is of the form: (i)T(A) = 0 for all A ∈ F ( X ) and (ii) T(A) = BA, for all A ∈ A ( X ) and some B ∈ L ( X ) .

Let R be a prime ring of characteristic different from 2, L be a non-central Lie ideal of R and m, n be the fixed positive integers. If R admits a generalized derivation F associated with a deviation d such that $$F(u^2)^m -d(u)^{2n}\in Z(R)$$ for all $$u \in L$$ , then R satisfies $$s_4$$ , the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

Let R be a prime ring, F be a generalized derivation associated with a derivation d of R and m, n be the fixed positive integers. In this paper we study the case when one of the following holds: (i) F(x) ◦m F(y) = (x ◦ y)n, (ii) F(x)◦md(y) = d(x◦y)n for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.

In this paper we prove that on a 2-torsion free semiprime ring R every Jordan triple (α,β)-derivation (resp. generalized Jordan triple (α,β)-derivation) on Lie ideal L is an (α,β)-derivation on L (resp. generalized (α,β)-derivation on L)

Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta,\varphi)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.

Let R be a semiprime ring of characteristic different from 2, C its extended centroid, Z(R) its center, F and G non-zero skew derivations of R with associated automorphism \(\alpha \) and m, n positive integers such that $$\begin{aligned}{}[F(x),G(y)]_m=[x,y]^n ~\mathrm{for \, all}~x,y \in R. \end{aligned}$$ Then R is commutative.

In this paper we prove the following result. Let R be a n!-torsion free semiprime ring and let f :R → R be an additive mapping satisfying the relation f(x)x n + x n f(x) = 0 for all x ∈ R. In this case f = 0.

Let R be a prime ring of characteristic different from 2 and m a fixed positive integer. If R admits a generalized derivation associated with a nonzero deviation d such that [F(x),d(y)] m =[x,y] for all x,y in some appropriate subset of R, then R is commutative. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer–Werner theorem.

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover we also examine the case when $R$ is a semiprime ring.

Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m, n ≥ 1 are fixed positive integers, F a generalized derivation associated with a nonzero derivation d of R. We study the case when one of the following holds: (i) F ( x ) ∘ m d ( y ) = ( x ∘ y ) n and (ii) ( F ( x ) ∘ d ( y ) ) m = ( x ∘ y ) n , for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring.

Let R be a semiprime ring with involution ∗ and let F, D : R → R be additive mappings satisfying the conditions (i) F(x 2) = F(x)x ∗+x ∗ D(x) and D(x 2) = D(x)x ∗+x ∗ D(x); (ii) F(x n + 1) = F(x)(x ∗) n +x ∗ D(x)(x ∗) n − 1+(x ∗)2 D(x)(x ∗) n − 2+⋯+(x ∗) n D(x) for all x ∈ R. Then, F(x y) = F(y)x ∗+y ∗ D(x) and D(x y) = D(y)x ∗+y ∗ D(x) for all x, y ∈ R.

In the present paper it is shown that if R is a semiprime ring of characteristic different from two which admits a derivation d such that \([d(x^m),d(y^n)]=\pm [x^m, y^n]\) for all \(x,y\in R\), where \(m\ge 1, n\ge 1\) are fixed positive integers, then R is commutative. Further using this result it is established that if \(\mathfrak {A}\) is a semiprime Banach algebra and \(\mathscr {H}_1\) and \(\mathscr {H}_2\) are nonvoid open subsets of \(\mathfrak {A}\) which admits a continuous derivation \(d:\mathfrak {A}\rightarrow \mathfrak {A}\) such that \([d(x^m),d(y^n)]\pm [x^m,y^n]=0\) for all \(x\in \mathscr {H}_1\) and \(y\in \mathscr {H}_2\), where m, n are no longer fixed rather they depend on the pair of elements \( x, y \in \mathfrak {A}\), then \(\mathfrak {A}\) is commutative.

The main purpose of this paper is to prove the following result: Let n> 1 be a fixed integer, let R be a n!-torsion free semiprime ring, and let f : R → R be an additive mapping satisfying the relation ( f( x), x)n =( ( ... (( f( x), x) ,x ) , . .. ) ,x )= 0 for all x ∈ R. In this case ( f( x), x )= 0 is fulfilled for all x ∈ R. Since any semisimple Banach algebra (for example, C ∗ algebra) is semiprime, this purely algebraic result might be of some interest from functional analysis point of view.

The aim of this paper is to prove the next result. Let n > 1be an integer and let R be a n!-torsion free semiprime ring. Suppose thatf : R →R is an additive mapping satisfying the relation [f(x),x n ] = 0for all x ∈R. Then f is commuting on R. 1. Introduction and the maintheoremThroughout, R will represent an associative ring with a center Z(R). Letn > 1 be an integer. A ring R is n-torsion free if nx = 0, x ∈ R, impliesx = 0. The Lie product (or a commutator) of elements x,y ∈ R will be denotedby [x,y] (i.e., [x,y] = xy − yx). Recall that a ring R is prime if aRb = {0},a,b ∈ R, implies that either a = 0 or b = 0. Furthermore, a ring R is calledsemiprime if aRa = {0}, a ∈ R, implies a = 0. We will denote by C and Q theextended centroid and the maximal right ring of quotients of a semiprime ringR, respectively. For the explanation of the extended centroid as well as themaximal right ring of quotients of a semiprime ring we refer the reader to [4].As usual, the socle of a ring R will be denoted by soc(R).An additive mapping D : R → R is called a derivation on R if D(xy) =D(x)y + xD(y) holds for all pairs x,y ∈ R. An additive mapping f : R →R is called centralizing on R if [f(x),x] ∈ Z(R) holds for all x ∈ R. In aspecial case, when [f(x),x] = 0 for all x ∈ R, the mapping f is said to becommuting on R. A classical result of Posner [21] (Posner’s second theorem)states that the existence of a nonzero centralizing derivation on a prime ringforces the ring to be commutative. Posner’s second theorem in general cannotbe proved for semiprime rings as shows the following example. Let R

Let R be an associative ring, and F : R ! R an additive mapping. The map F is called a Jordan triple derivation if F (xyx) = F (x)yx + xF (y)x + xyF (x) for all x, y 2 R which is fulfilled (2), and F is called a generalized Jordan triple derivation if F (xyx) = F (x)yx + xf (y)x + xyf (x) with some Jordan triple derivation f for all x, y 2 R which is fulfilled (13). In this note, we deal with generalized Jordan triple derivations of semiprime rings, and give an affirmative answer to our conjecture in (12).

In this paper, functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let \({\mathcal {L}}(X)\) be the algebra of all bounded linear operators of X into itself and let \({\mathcal {A}}(X)\subset {\mathcal {L}}(X)\) be a standard operator algebra. Suppose there exist linear mappings \(D, G:{\mathcal {A}}(X)\rightarrow {\mathcal {L}}(X)\) satisfying the relations \(D(A^{2n+1})=D(A^{2n})A+A^{2n}G(A)\) and \(G(A^{2n+1})=G(A^{2n})A+A^{2n}D(A)\) for all \(A\in {\mathcal {A}}(X)\). Then there exists \(B\in {\mathcal {L}}(X)\) such that \(D(A)=G(A)=[A,B]\) for all \(A\in {\mathcal {A}}(X)\).

The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained.

Let R be a 2-torsion free ring and L be a Lie ideal of R. In this paper we initiate the study of generalized derivations on Lie ideals associate with Hochschild 2-cocycles and prove that every generalized derivation associate with a Hochschild 2-cocycle on L is a generalized derivation on L under certain conditions.

In [2, Theorem 3], Bell and Kappe proved that if d is a derivation of a prime ring R which acts as a homomorphism or an anti-homomorphism on a nonzero right ideal I of R, then d = 0 on R. In the present paper our objective is to extend this result to Lie ideals. The following result is proved: Let R be a 2-torsion free prime ring and U a nonzero Lie ideal of R such that u2 ∈ U, for all u ∈ U. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U, then either d=0 or U ⫅Z(R).

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.

Let R be a ring and F and G be generalized derivations of R with associated derivations d and g respectively. In the present paper, we shall investigate the commutativity of semiprime ring R admitting generalized derivations F and G satisfying any one of the properties: (i)F(x)x = xG(x); (ii) [F(x); d(y)] = [x; y]; (iii) F([x; y]) = [d(x); F(y)]; (iv) d(x)F(y) = xy; (v) F(x2) = x2; (vi) [F(x); y] = [x;G(y)]; (vii) F([x; y]) = [F(x); y]+[d(y); x] and (viii) F(x ◦ y) = F(x) ◦ y − d(y) ◦ x for all x; y in some appropriate subset of R.