We prove that in a ring Rwith an identity there exists an element a ∈ R and a nonzero derivation d ∈ Der R such that ad(a) 6= 0. A ring R is said to be a d-rigid ring for some derivation d ∈ Der R if d(a) = 0 or ad(a) 6= 0 for all a ∈ R. We study rings with rigid derivations and establish that a commutative Artinian ring R either has a non-rigid derivation or R = R1 ⊕ · · · ⊕ Rn is a ring direc...

let a be a unital r-algebra and m be a unital a-bimodule. it is shown that every jordan derivation of the trivial extension of a by m, under some conditions, is the sum of a derivation and an antiderivation.

Let n ∈ N − {1}, and let A be a Banach algebra. An additive map D : A → A is called n-Jordan derivation if D(a) = D(a)a + aD(a)a + ...+ aD(a)a+ aD(a), for all a ∈ A. Using fixed point methods, we investigate the stability of n–Jordan derivations (n–Jordan ∗−derivations) on Banach algebras (C∗−algebras). Also we show that to each approximate ∗−Jordan derivation f in a C∗− algebra there correspon...

In this paper we introduce derivations in Krasner hyperrings and derive some basic properties of derivations. We also prove that for a strongly differential hyperring $R$ and for any strongly differential hyperideal $I$ of $R,$ the factor hyperring $R/I$ is a strongly differential hyperring. Further we prove that a map $d: R rightarrow R$ is a derivation of a hyperring $R$ if and only if the in...

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that usH(u)ut = 0 for all u ∈ L, where s ≥ 0, t ≥ 0 are fixed integers. Then H(x) = 0 for all x ∈ R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x, y ∈ R, the commutator xy− yx will be denoted by [x, y]. An add...

In this paper we prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let A(X) be a standard operator algebra on X and let L(X) be an algebra of all bounded linear operators on X. Suppose we have a linear mapping D : A(X) → L(X) satisfying the relation D(Am+n) = D(Am)An + AmD(An) for all A ∈ A(X) and some fixed integers m ≥ 1, ...

We prove in this note the following result. Let n > 1 be an integer and let R be an n!torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D : R→ R such that D(xn) =∑nj=1 xn− jD(x)x j−1is fulfilled for all x ∈ R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associ...

Let A be a C-algebra acting on a Hilbert space H, σ : A → B(H) be a linear mapping and d : A → B(H) be a σ-derivation. Generalizing the celebrated theorem of Sakai, we prove that if σ is a continuous ∗-mapping then d is automatically continuous. In addition, we show the converse is true in the sense that if d is a continuous ∗-σ-derivation then there exists a continuous linear mapping Σ : A → B...

Let A be a C-algebra acting on a Hilbert space H, σ : A → B(H) be a linear mapping and d : A → B(H) be a σ-derivation. Generalizing the celebrated theorem of Sakai, we prove that if σ is a continuous ∗-mapping then d is automatically continuous. In addition, we show the converse is true in the sense that if d is a continuous ∗-σ-derivation then there exists a continuous linear mapping Σ : A → B...