In this note, by means of the spectrum of the generating operator, we characterize the self-adjointness and closedness of the range of a normal and a self-adjoint Jordan *-derivation, respectively.

We show that a quaternary Jordan derivation on a quaternary Banach algebra associated with the equation f( x+ y + z 4 ) + f( 3x− y − 4z 4 ) + f( 4x+ 3z 4 ) = 2f(x) . is satisfied in generalized Hyers–Ulam stability.

Let M be a 2-torsion free prime Γ-ring and X a nonzero faithful and prime ΓM -module. Then the existence of a nonzero Jordan left derivation d : M → X satisfying some appropriate conditions implies M is commutative. M is also commutative in the case that d : M → M is a derivation along with some suitable assumptions. AMS (MOS) Subject Classification Codes: 03E72, 54A40, 54B15

Let $A$ be an unital alternative $*$-algebra. Assume that contains a nontrivial symmetric idempotent element $e$ which satisfies $xA \cdot e = 0$ implies $x and (1_A - e) 0$. In this paper, it is shown $\Phi$ nonlinear $*$-Jordan-type derivation on A if only additive $*$-derivation. As application, we get result $W^{*}$-algebras.

Let R be a ring and S a nonempty subset of R. Suppose that θ and φ are endomorphisms of R. An additive mapping δ : R → R is called a left (θ,φ)-derivation (resp., Jordan left (θ,φ)derivation) on S if δ(xy) = θ(x)δ(y)+φ(y)δ(x) (resp., δ(x2) = θ(x)δ(x)+φ(x)δ(x)) holds for all x,y ∈ S. Suppose that J is a Jordan ideal and a subring of a 2-torsion-free prime ring R. In the present paper, it is show...

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, ...

In the present paper, we study simple algebras, which do not belong to well-known classes of algebras (associative alternative Lie Jordan etc.). The finite-dimensional over a field characteristic 0 without finite basis identities, constructed by Kislitsin, are such algebras. consider two algebras: seven-dimensional anticommutative algebra \(\mathcal{D}\) and central commutative \(\mathcal{C}\)....

Let m ≥ 0, n ≥ 0 be fixed integers with m + n 6= 0 and let R be a ring. It is our aim in this paper to investigate additive mapping T : R → R satisfying the relation (m + n)T (x2) = mT (x)x + nxT (x) for all x ∈ R. This research is a continuation of our earlier work ([11]). Throughout, R will represent an associative ring with center Z(R). Given an integer n ≥ 2, a ring R is said to be n−torsio...