Derivations and generalized derivations

The present paper deals with the commutativity of an associative ring R and a unital Banach Algebra A via derivations. Precisely, the study of multiplicative (generalized)-derivations F and G of semiprime (prime) ring R satisfying the identities G(xy) ± [F (x), y] ± [x, y] ∈ Z(R) and G(xy) ± [x, F (y)] ± [x, y] ∈ Z(R) has been carried out. Moreover, we prove that a unital prime Banach algebra A admitting continuous linear generalized derivations F and G is commutative if for any integer n > 1 either G((xy) n) + [F (x n), y n ] + [x n , y n ] ∈ Z(A) or G((xy) n) − [F (x n), y n ] − [x n , y n ] ∈ Z(A).

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

Let N be a prime left near-ring with multiplicative centerZ; and D be a (α, γ)derivation such that δD = Dδ and ΓD = DΓ(i)If D(N)⊂ Z; or [D(N);D(N)] = 0 or [D(N);D(N)]σ, γ= 0; then (N; +)is abelian. (ii) If N is 2-torsion free, d1 is a (α, γ)-derivation and d2 is a derivation on N such that d1d2(N) = 0, then d1 = 0 or d2 = 0. Abstract. Let N be a prime left near-ring with multiplicative center Z, and D be a (σ, τ )-derivation such that σD = Dσ and τ D = Dτ. (i) If D(N ) ⊂ Z, or [D(N ), D(N )] = 0 or [D(N ), D(N )]σ,τ = 0, then (N, +) is abelian. (ii) If N is 2-torsion free, d1 is a (σ, τ )-derivation and d2 is a derivation on N such that d1d2(N ) = 0, then d1 = 0 or d2 = 0.

In the present paper we prove the following result; Let R be a noncommutative prime ring, I an ideal of R, (F, d) a generalized derivation of R and a ∈ R. If F ([x, a]) = 0 or [F (x), a] = 0 for all x ∈ I, then, d(x) = λ[x, a] for all x ∈ I or a ∈ Z.

In this paper, we define a set including of all fa with a ∈ R generalized derivations of R and is denoted by f R. It is proved that (i) the mapping g : L (R) → f R given by g (a) = f −a for all a ∈ R is a Lie epimorphism with kernel Nσ,τ ; (ii) if R is a semiprime ring and σ is an epimorphism of R, the mapping h : f R → I (R) given by h (fa) = i σ(−a) is a Lie epimorphism with kernel l (f R) ; (iii) if f R is a prime Lie ring and A, B are Lie ideals of R, then [f A , f B ] = (0) implies that either f A = (0) or f B = (0).

Let R be a prime ring with a characteristic not equal to two, σ, τ be automorphisms of R, and d be a nonzero derivation of R commuting with σ and τ. It is proved that for any (σ, τ)-left Lie ideal U of R: (1) if d(U) ⊆ Z, then σ(u) + τ (u) ∈ Z, for all u ∈ U , (2) if d 2 (U) = 0, then σ(u) + τ (u) ∈ Z, for all u ∈ U , (3) if char R 6 = 2, 3, d(U) ⊆ U and d 2 (U) ⊆ Z, then σ(u) + τ (u) ∈ Z, for all u ∈ U .

In the present paper we prove the following result; Let R be a noncommutative prime ring, I an ideal of R, (F, d) a generalized derivation of R and a ∈ R. If F ([x, a]) = 0 or [F (x), a] = 0 for all x ∈ I, then, d(x) = λ[x, a] for all x ∈ I or a ∈ Z.

Let R be a -prime ring with characteristic not 2; Z(R) be the center of
R; I be a nonzero -ideal of R; ;  : R ! R be two automorphisms, d
be a nonzero (; )-derivation of R and h be a nonzero derivation of R:
In the present paper, it is shown that (i) If d (I)  C; and  commutes
with  then R is commutative. (ii) Let  and  commute with : If
a 2 I \ S (R) and [d(I); a];  C; then a 2 Z(R): (iii) Let ; 
and h commute with : If dh (I)  C; and h (I)  I then R is
commutative.

Siberian Mathematical Journal, Vol. 48, No. 6, pp. 979–983, 2007 Original Russian Text Copyright c 2007 Gölbasi ¨O. and Aydin N. ... ORTHOGONAL GENERALIZED (σ, τ)-DERIVATIONS OF SEMIPRIME RINGS ... Abstract: This paper abstracts some results of M. Bresar and J. ...

Assume (1-5) : 1 Y = f(X) 2 Y = alpha + beta X 3 Y = alpha + beta X + u 4 Y = f(X,u) 5 Y = alpha + beta X subi + u subi i=1,2,...,n E(u subi) = 0 for all I E(u subi u subj) = { 0, sigma subu^2 } for i not equal j ; i,j =1,2, …,n for i equal j ; i,j = 1,2, …,n Arithmetic Means Xbar = 1/n Sum{i=1,n} X subi Ybar = 1/n Sum{i=1,n} Y subi 6 Y hat = alpha hat + beta hat X The principle of least squares is that the alpha hat, beta hat values should be chosen so as to make Sum e^2 as small as possible. (p. 9) A necessary condition is that the partial derivatives of the sum with respect to alpha hat and beta hat should both be zero. Sum{i=1,n} e^2 = f(alpha hat, beta hat) 7 Sum{i=1,n} Y subi = n alpha hat + beta hat Sum{i=1,n} X subi Sum {i=1,n} X subi Y subi = alpha hat Sum{i=1,n} X subi + beta hat Sum{i=1,n} X subi^2

Let R be a σ−prime ring. An additive mapping F : R → R is called a generalized (α, α) − derivation, if there exists a mapping g : R → R such that F (xy) = F (x)α(y) + α (x) g(y) for all x, y ∈ R. In this paper, some results about generalized derivation are extended to generalized (α, α) −derivation in σ−prime ring.

Let R be a prime ring with characteristic not two. U a (σ, τ)-left Lie ideal of R and d : R → R a non-zero derivation. The purpose of this paper is to invesitigate identities satisfied on prime rings. We prove the following results: (1) [d(R), a] = 0 ⇔ d([R, a]) = 0. (2) if (R, a) σ,τ = 0 then a ∈ Z. (3) if (R, a) σ,τ ⊂ C σ,τ then a ∈ Z. (4) if (U, a) ⊂ Z then a 2 ∈ Z or σ(u) + τ (u) ∈ Z, for all u ∈ U. (5) if (U, R) σ,τ ⊂ C σ,τ then U ⊂ Z.

Let R be a prime ring of characteristic di®erent from two, d : R ! R a nonzero derivation, and M a non-zero left ideal of R: We prove the following results: (1) if a 2 R and [d(R); a]¾ ;¿ = 0; then ¾ (a) + ¿(a) 2 Z; the center of R; (2) if d([R; a] ¾ ;¿ ) = 0; then ¾ (a) + ¿(a) 2 Z; (3) if ([R; M ] ¾ ;¿ ; a) ¾ ;¿ = 0; then a 2 Z; (4) (d(R); a) = 0 if, and only if, d((R; a)) = 0: