Let R be a prime ring of characteristic different from 2 and 3, Qr its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y \in R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y \in R. We prove that, if F is a nonzero generalized skew derivation of R such that F(x)×[F(x), x]n = 0 for any x \in L, then either there exists λ \in C such that F(x) = λx for all x \in R, or R\subset M_{2}\left ( C \right ) and there exist a \in Qr and λ \in C such that F(x) = ax + xa + λx for any x \in R., Vincenzo De Filippis., and Obsahuje seznam literatury
We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra $\mathcal A$. We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice $\mathcal D (A)$ of all deductive systems on $\mathcal A$. Moreover, relative annihilators of $C\in \mathcal D (A)$ with respect to $B \in \mathcal D (A)$ are introduced and serve as relative pseudocomplements of $C$ w.r.t. $B$ in $\mathcal D (A)$.
The concepts of an annihilator and a relative annihilator in an autometrized $l$-algebra are introduced. It is shown that every relative annihilator in a normal autometrized $l$-algebra $\mathcal {A}$ is an ideal of $\mathcal {A}$ and every principal ideal of $\mathcal {A}$ is an annihilator of $\mathcal {A}$. The set of all annihilators of $\mathcal {A}$ forms a complete lattice. The concept of an $I$-polar is introduced for every ideal $I$ of $\mathcal {A}$. The set of all $I$-polars is a complete lattice which becomes a two-element chain provided $I$ is prime. The $I$-polars are characterized as pseudocomplements in the lattice of all ideals of $\mathcal {A}$ containing $I$.