Derivations in Rings and Algebras

Let R be a ring and α, β be automorphisms of R. An additive mapping F : R → R is called a generalized (α, β)-derivation on R if there exists an (α, β)derivation d: R → R such that F (xy) = F (x)α(y) + β(x)d(y) holds for all x, y ∈ R. For any x, y ∈ R, set [x, y] α,β = xα(y) − β(y)x and (x • y) α,β = xα(y) + β(y)x. In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized (α, β)-derivations F and G satisfying any one of the following properties: (i) F ([x, y]) = (x • y) α,β , (ii) F (x • y) = [x, y] α,β , (iii) [F (x), y] α,β = (F (x) • y) α,β , (iv) F ([x, y]) = [F (x), y] α,β , (v) F (x • y) = (F (x) • y) α,β , (vi) F ([x, y] = [α(x), G(y)] and (vii) F (x • y) = (α(x) • G(y)) for all x, y in some appropriate subset of R. Finally, obtain some results on semi-projective Morita context with generalized (α, β)-derivations.

Let N be a 2-torsion free s-prime near-ring and 1, a Î Aut N. Let d be a nonzero (1, a)-derivation which commutes with s, and L be a nonzeros-Lie ideal, then N is commutative if any one of the following conditions hold. (i) [d(u), u] 1,a = 0. (ii) d(u)u = a(u)d(u) (iii) d(u 2) = ± a(u 2) (iv) d(u 2) = 2 d(u)a(u), for all u Î L.

In this article, we introduce new generators of a permuting n-derivations to improve and increase the action of usual derivation. We produce a permuting n-generalized semiderivation, a permuting n-semigeneralized semiderivation, a permuting n-antisemigeneralized semiderivation and a permuting skew n-antisemigeneralized semiderivation of non-empty rings with their applications. Actually, we study the behaviour of those types and present their results of semiprime ring R. Examples of various results have also been included. That is, many of the branches of science such as business, engineering and quantum physics, which used a derivation, have the opportunity to invest them in solving their problems.

Let M be a noncommutative 2-torsion free semiprime Γ-ring satisfying a certain assumption and let S and T be left centralizers on M. We prove the following results: (i) If [S(x), T (x)] α βS(x) + S(x)β[S(x), T (x)] α =0 holds for all x ∈ M and α, β ∈ Γ, then [S(x), T (x)] α =0. (ii) If S = 0(T = 0), then there exists λ ∈ C,(the extended centroid of M) such that T =λαS (S = λαT) for all α ∈ Γ. (iii) Suppose that [[S(x), T (x)] α , S(x)] β =0 holds for all x ∈ M and α, β ∈ Γ. Then [S(x), T (x)] α =0 for all x ∈ M and α ∈ Γ. (iv) If M is a prime Γ-ring satisfying a certain assumption and S = 0(T = 0), then there exists λ ∈ C, the extended centroid, such that T =λαS(S = λαT).

Let $M$ be a noncommutative 2-torsion free semiprime $\Gamma$-ring satisfying a certain assumption and let $S$ and $T$ be left centralizers on $M$. We prove the following results: \\(i) If $[S(x),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]_{\alpha }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in \Gamma $, then $[S(x),T(x)]_{\alpha }$=$0$. \\(ii) If $S\neq 0 (T\neq 0)$, then there exists $\lambda \in C$,(the extended centroid of $M$) such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$ for all $\alpha \in \Gamma $. \\(iii) Suppose that $[[S(x),T(x)]_{\alpha },S(x)]_{\beta }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in\Gamma $. Then $[S(x),T(x)]_{\alpha }$=$0$ for all $x\in M$ and $\alpha \in\Gamma $. \\(iv) If $M$ is a prime $\Gamma $-ring satisfying a certain assumption and $S\neq 0(T\neq 0)$, then there exists $\lambda \in C$, the extended centroid, such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$.