Let A be a Banach algebra and M be a Banach A-bimodule. We say that a linear mapping d: A®M is a generalized s-derivation whenever there exists a s-derivation d: A ® M such that d(ab) =d(a) s(b) +s(a)d(b), for all a, bÎ A. Giving some facts concerning generalized s-derivations, we prove that if A is unital and if d: A®A is a generalized s-derivation and there exists an elementa 2 A such that d (a) is invertible, then d is continuous if and only if d is continuous. We also show that if M is unital, has no zero divisor and d: A® M is a generalized s-derivation such that d(1) ¹0, then ker (d) is a bi-ideal of A and ker(d) =ker (s) =ker (d), where 1 denotes the unit element of A.

Our intention in this paper is to prove the following. Let R be a ring with an idempotent element (0, 1 ̸=)e and f be a multiplicative b-generalized derivation on R. Then we show that f is additive byimposing certain conditions on the ring R.

Let R be a prime ring with extended centroid C, H a generalized derivation of R and n³1 a fixed integer. In this paper we study the situations: (1) If (H (xy)) n= (H (x)) n (H (y)) n for all x, y Î R; (2) obtain some related result in case R is a noncommutative Banach algebra and H is continuous or spectrally bounded.

Abstract. In this paper, we study the notion of semi-derivation in Krasner hyperring and present some examples of them. We introduce the concept of generalized semi-derivation in Krasner hyperring and present some examples. Then, we derive some properties of semi-derivation on Krasner hyperring which proves the commutativity of a Krasner hyperring. Later we prove if f is a non-zero semi-derivation on Krasner hyperring R, then f 2 6= 0 on R. Finally, for a generalized semi-derivation F on R, if F(u ◦ v) = 0, for all u, v ∈ I, then R is commutative.

We know that every Jordan generalized k-derivation of a é-ring is not a generalized k-derivation of the same, in general. Here, we develop a number of lemmas relating to these derivations of certain é-rings and we show that under some conditions every Jordan generalized k-derivation of a 2-torsion free semiprime é-N ring is a generalized k-derivation.

Let $R$ be a prime ring and $Z(R)$ denotes the center of $R$. In this study, we expose the commutativity of $R$ as a consequence of specific differential identities involving derivations acting on left ideals of $R$. Finally, we give examples that demonstrate the necessity of hypotheses taken in the theorems.

A prime ring ${S}$ with the centre ${Z}$ and generalised derivations that meet certain algebraic identities is considered. Let's assume that $\Psi$ and $\Phi$ are two generalised derivations associated with $\psi$ and $\phi$ on ${S},$ respectively. In this article, we examine the following identities: (i) $\Psi(a)b-a\Phi(b)\in {Z},$ (ii) $\Psi(a)b-b\Phi(a)\in {Z},$ (iii) $\Psi(a)a-b\Phi(b)\in {Z},$ (iv) $\Psi(a)a-a\Phi(b)\in {Z},$ (v) $\Psi(a)a-b\Phi(a)\in {Z},$ for every $a, b\in {J},$ where ${J}$ is a non-zero two sided ideal of ${S}.$ We also provide an example to show that the condition of primeness imposed in the hypotheses of our results is essential.