Let R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R.If [G(u), u](1) R satisfies the standard identity s(2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.
This paper aims to study spacelike surfaces from a given spacelike curve in Minkowski 3–space. Also, we investigate the necessary and sufficient conditions for the given space-like curve to be the line of curvature on the space-like surface. Depending on the causal character of the curve, the necessary and sufficient conditions for the given space-like curve to satisfy the line of curvature and the geodesic (resp. asymptotic) requirements are also analyzed. Furthermore, we give with illustration some computational examples in support of our main results.
Let be a 2-torsion free ring and let be a noncentral Lie ideal of , and let : → and : → be two generalized derivations of . We will analyse the structure of in the following cases: (a) is prime and ( ) = ( ) for all ∈ and fixed positive integers ̸ = ; (b) is prime and (( V ) ) = ((V ) ) for all , V ∈ and fixed integers , , , , , ≥ 1; (c) is semiprime and (( V) ) = ((V ) ) for all , V ∈ [ , ] and fixed integer ≥ 1; and (d) is semiprime and (( V) ) = ((V ) ) for all , V ∈ and fixed integer ≥ 1.
If F, D : R → R are additive mappings which satisfyThen, F is a generalized left derivation with associated Jordan left derivation D on R.Similar type of result has been done for the other identity forcing to generalized derivation and at last an example has given in support of the theorems.
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