Let $R$ be a prime ring, with no non-zero nil right ideal, $d$ a non-zero drivation of $R$, $I$ a non-zero two-sided ideal of $R$. If, for any $x$, $y\in I$, there exists $n=n\left(x,y\right)\ge 1$ such that ${\left(d\left(\left[x,y\right]\right)-\left[x,y\right]\right)}^n=0$, then $R$ is commutative. As a consequence we extend the result to Lie ideals.

Under some conditions we prove that every generalized Jordan triple derivation on a Lie triple system is a generalized derivation. Specially, we conclude that every Jordan triple $\theta$-derivation on a Lie triple system is a $\theta$-derivation.

Let $R$ be a $2$-torsion free $*$-prime ring, $d$ a derivation which commutes with $*$ and $J$ a $*$-Jordan ideal and a subring of $R$. In this paper, it is shown that if either $d$ acts as a homomorphism or as an anti-homomorphism on $J$, then $d=0$ or $J\subseteq Z\left(R\right)$. Furthermore, an example is given to demonstrate that the $*$-primeness hypothesis is not superfluous.

Let $R$ be a $2$-torsion free prime ring. Suppose that $\theta ,\phi$ are automorphisms of $R$. In the present paper it is established that if $R$ admits a nonzero Jordan left $(\theta ,\theta )$-derivation, then $R$ is commutative. Further, as an application of this resul it is shown that every Jordan left $(\theta ,\theta )$-derivation on $R$ is a left $(\theta ,\theta )$-derivation on $R$. Finally, in case of an arbitrary prime ring it is proved that if $R$ admits a left $(\theta ,\phi )$-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of $R$, then $d=0$...

Let $R$ be a 2-torsion free prime ring and let $U$ be a Lie ideal of $R$ such that $u^2\in U$ for all $u\in U$. In the present paper it is shown that if $d$ is an additive mappings of $R$ into itself satisfying $d\left(u^2\right)=2ud\left(u\right)$ for all $u\in U$, then $d\left(uv\right)=ud\left(v\right)+vd\left(u\right)$ for all $u,v\in U$.

Let k be a field. We prove that any polynomial ring over k is a Kadison algebra if and only if k is infinite. Moreover, we present some new examples of Kadison algebras and examples of algebras which are not Kadison algebras.

Let $N$ be a $3$-prime left near-ring with multiplicative center $Z$, a $(\sigma ,\tau )$-derivation $D$ on $N$ is defined to be an additive endomorphism satisfying the product rule $D\left(xy\right)=\tau \left(x\right)D\left(y\right)+D\left(x\right)\sigma \left(y\right)$ for all $x,y\in N$, where $\sigma$ and $\tau$ are automorphisms of $N$. A nonempty subset $U$ of $N$ will be called a semigroup right ideal (resp. semigroup left ideal) if $UN\subset U$ (resp. $NU\subset U$) and if $U$ is both a semigroup right ideal and a semigroup left ideal, it be called a semigroup ideal. We prove the following results: Let $D$ be a $(\sigma ,$