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.

Let $𝒦$ be a semiprime ring and $T:𝒦\to 𝒦$ an additive mapping such that $T\left(x^2\right)=T\left(x\right)x$ holds for all $x\in 𝒦$. Then $T$ is a left centralizer of $𝒦$. It is also proved that Jordan centralizers and centralizers of $𝒦$ coincide.

Let F be a commutative ring with unit. In this paper, for an associative F-algebra A we study some properties forced by finite length or DCC condition on F-submodules of A that are subalgebras with zero multiplication. Such conditions were considered earlier when F was either a field or the ring of rational integers. In the final section, we consider algebras with maximal commutative subalgebras of finite length as F-modules and obtain some results parallel to those known for ACC condition or finite...

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 $u,v\in U$.