Let 𝒜 be a prime ∗-algebra. For any A , B ∈ A , a product A ⋆ B = A ⁢ B * + B ⁢ A * is called a bi-skew Jordan product. In this paper, it is shown that every multiplicative bi-skew Jordan triple derivation is an additive ∗-derivation on a prime ∗-algebra.

The aim of this paper is to discuss the commutativity of a Banach algebra $A$ via its derivations. In particular, we prove that if $A$ is a unital prime Banach algebra and $A$ has a nonzero continuous linear derivation $d:A\rightarrow A$ such that either $d((xy)^{m})-x^{m}y^{m}$ or $d((xy)^{m})-y^{m}x^{m}$ is in the centre of $A$ for an integer $m=m(x,y)$ and sufficiently many $x,y$ , then $A$ is commutative. We give examples to illustrate the scope of the main results and show that the hypotheses are not superfluous.

Let R be a noncommutative prime ring, U be the left Utumi quotient ring of R, and k, m, n, r be fixed positive integers. If there exist a generalized derivation G and a derivation g (which is independent of G) of R such that [G(x^m)xⁿ + xⁿg(x^m), x^r]_k = 0, for all x ∈ R, then there exists a ∈ U such that G(x) = ax, for all x ∈ R. As a consequence of the result in the present article, one may obtain Theorem 1 in Demir and Argaç [ 10 Demir, Ç. , Argaç, N. (2010). A result on generalized derivations with Engel conditions on one-sided ideals. J. Korean Math. Soc. 47(3):483–494. [Crossref], [Web of Science ®] [Google Scholar] ].

Let R be an associative ring. An additive map x -> x* of R into itself is called an involution if (i) (xy* = y *x* and (ii) (x*)* = x hold for all x is an element of R. The main purpose of this paper is to study some additive mappings on prime and semiprime rings with involution. Moreover, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the various results are not superfluous.

In this article, we investigate the behaviour of a non-linear map ? on *-algebra A, which satisfies ?([X?Y,Z]?) = [?(X)?Y,Z]? +[X??(Y),Z]? +[X?Y,?(Z)]?, where X?Y = XY+YX and [X,Y]? = XY* ?YX* (namely, Jordan and bi-skewLie product, respectively), for all X,Y,Z ? A. Furthermore, we apply the above mentioned result to several distinct algebras.