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
Whereas most plant suspension cultures are grown heterotrophically in the presence of sugars, a limited number of photoautotrophic cultures have been established which are able to grow with CO2 as the sole carbon source. Photoautotrophic cultures are useful to address various aspects of photosynthesis, source-sink regulation, nitrogen metabolism, production of secondary metabolites, and defence responses. The homogenous populations of these cultures provide an ideal and sensitive system to obtain reproducible results. The availability of an increasing number of photoautotrophic cultures from different economically important species provides the basis also for practical applications. and T. Roitsch, A. K. Sinha.