Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m,n$ fixed positive integers. (i) If ${\left(d[x,y]\right)}^m={[x,y]}_n$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathrm{Char}R\ne 2$ and ${[d\left(x\right),d\left(y\right)]}_m={[x,y]}^n$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

Let $R$ be a prime ring of char $R\ne 2$ with a nonzero derivation $d$ and let $U$ be its noncentral Lie ideal. If for some fixed integers $n_1\ge 0,n_2\ge 0,n_3\ge 0$, ${\left(u^{n_1}[d\left(u\right),u]u^{n_2}\right)}^{n_3}\in Z\left(R\right)$ for all $u\in U$, then $R$ satisfies $S_4$, the standard identity in four variables.

We characterize left Noetherian rings which have only trivial derivations.

We study effectively the Cartan geometry of Levi-nondegenerate C 6-smooth hypersurfaces M 3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M 3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.

If $\tau$ is a hereditary torsion theory on ${\mathrm{𝐌𝐨𝐝}}_R$ and $Q_{\tau }:{\mathrm{𝐌𝐨𝐝}}_R\to {\mathrm{𝐌𝐨𝐝}}_R$ is the localization functor, then we show that every $f$-derivation $d:M\to N$ has a unique extension to an $f_{\tau }$-derivation $d_{\tau }:Q_{\tau }\left(M\right)\to Q_{\tau }\left(N\right)$ when $\tau$ is a differential torsion theory on ${\mathrm{𝐌𝐨𝐝}}_R$. Dually, it is shown that if $\tau$ is cohereditary and $C_{\tau }:{\mathrm{𝐌𝐨𝐝}}_R\to {\mathrm{𝐌𝐨𝐝}}_R$ is the colocalization functor, then every $f$-derivation $d:M\to N$ can be lifted uniquely to an $f_{\tau }$-derivation $d_{\tau }:C_{\tau }\left(M\right)\to C_{\tau }\left(N\right)$.

The purpose of this note is to show how calculi on unital associative algebra with universal right bimodule generalize previously studied constructions by Pusz and Woronowicz [1989] and by Wess and Zumino [1990] and that in this language results are in a natural context, are easier to describe and handle. As a by-product we obtain intrinsic, coordinate-free and basis-independent generalization of the first order noncommutative differential calculi with partial derivatives.

We identify some situations where mappings related to left centralizers, derivations and generalized $(\alpha ,\beta )$-derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation $T$, of a semiprime ring $R$ the mapping $\psi R\to R$ defined by $\psi \left(x\right)=T\left(x\right)x-xT\left(x\right)$ for all $x\in R$ is a free action. We also show that for a generalized $(\alpha ,\beta )$-derivation $F$ of a semiprime ring $R,$ with associated $(\alpha ,\beta )$-derivation $d,$ a dependent element $a$ of $F$ is also a dependent element of $\alpha +d.$ Furthermore, we prove that for a centralizer $f$ and...

We investigate a new type of generalized derivations associated with Hochschild 2-cocycles which was introduced by A. Nakajima. We show that every generalized Jordan derivation of this type from CSL algebras or von Neumann algebras into themselves is a generalized derivation under some reasonable conditions. We also study generalized derivable mappings at zero point associated with Hochschild 2-cocycles on CSL algebras.

Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F\left(u\right){[F\left(u\right),u]}^n=0$ for all $u\in L$, where $n\ge 1$ is a fixed integer. Then one of the following holds: (1) ...

Let $𝔄=\left(\begin{array}{cc}𝒜& ℳ\\ & ℬ\end{array}\right)$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $(𝒜,ℬ)$-bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $[𝔏,𝔄]\subseteq 𝔏$. A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$, every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$.