in this paper, we classify the indecomposable non-nilpotent solvable lie algebras with $n(r_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $n(r_n,m,r)$.we also prove that these solvable lie algebras are complete and unique, up to isomorphism.

out, R will represent an associative ring with center Z(R). As usual the commutator xy−yx will be denoted by [x,y]. We will use basic commutator identities [xy,z] = [x,z]y+x[y,z] and [x,yz]= [x,y]z+y[x,z]. Recall that a ring R is prime if aRb = (0) implies that a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping x x∗ on a ring R is called involution in case (xy)∗ = ...

An all-fiber-ring resonator has been constructed using a single strand of single-mode optical fiber and a directional coupler. Derivation of the resonator finesse in terms of fiber and directional coupler parameters is given. A finesse of 80 has been achieved experimentally. Applications of such a fiber-ring resonator are discussed.

Let $R$ be a ring and $M$ a right $R$-module. We call $M$, coretractable relative to $overline{Z}(M)$ (for short, $overline{Z}(M)$-coretractable) provided that, for every proper submodule $N$ of $M$ containing $overline{Z}(M)$, there is a nonzero homomorphism $f:dfrac{M}{N}rightarrow M$. We investigate some conditions under which the two concepts coretractable and $overline{Z}(M)$-coretractable...

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds: 1. there exist α ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = αx, for all x ∈ R; 2. G is an usual derivatio...

The structures and properties of [n]sila-acenes (n=2-4) were investigated by density functional theory method. The results of calculations were obtained at B3LYP/6-311G (d,p) level on model species. Energetic criteria suggest that 2-1b (n=2), 3-1b (n=3), and 4-1b (n=4) isomers enjoy stabilization. By frontier orbital analysis, these systems are among the most stable of the family. Also, calcula...

let a be a unital r-algebra and m be a unital a-bimodule. it is shown that every jordan derivation of the trivial extension of a by m, under some conditions, is the sum of a derivation and an antiderivation.

The purpose of the present paper is to classify generalized derivations satisfying more specific algebraic identities in a prime ring with
 involution second kind. Some well-known results
 characterizing commutativity rings by have
 been using derivation.