Derivations as homomorphisms or anti-homomorphisms in differentially semiprime rings

Tensors as module homomorphisms over group rings

Braman [1] described a construction where third-order tensors are exactly the set of linear transformations acting on the set of matrices with vectors as scalars. This extends the familiar notion that matrices form the set of all linear transformations over vectors with real-valued scalars. This result is based upon a circulant-based tensor multiplication due to Kilmer et al. [4]. In this work,...

Derivations in semiprime rings and Banach algebras

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,yin R$ where $0neq bin R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. ...

A note on derivations in semiprime rings

We prove in this note the following result. Let n > 1 be an integer and let R be an n!torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D : R→ R such that D(xn) =∑nj=1 xn− jD(x)x j−1is fulfilled for all x ∈ R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associ...

On (θ, θ)-Derivations in Semiprime Rings

Generalized Derivations in Semiprime Gamma Rings