On ( θ , θ )-Derivations in Semiprime Rings

Mathematica Slovaca, 2012

The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained.

Boletim da Sociedade Paranaense de Matemática, 2010

The purpose of this note is to prove the following result. Let R be a semiprime ring of characteristic not 2 and G: R-→R be an additive mapping such that G(x 2 ) = G(x)x + xD(x) holds for all x ∈ R and some derivations D of R. Then G is a Jordan generalized derivation.

Turkish Journal of Mathematics, 2011

We investigate some properties of generalized (α, β)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (α, β)-derivation, with associated (α, β)-derivation δ , on a semiprime ring R such that [g(x), α(x)] = 0 for all x ∈ R , then δ(x)[y, z] = 0 for all x, y, z ∈ R and δ is central. We also show that if α, ν, τ are endomorphisms and β, μ are automorphisms of a semiprime ring R and if R has a generalized (α, β)-derivation g , with associated (α, β)-derivation δ , such that g([μ(x), w(y)]) = [ν(x), w(y)]α,τ , where w : R → R is commutativity preserving, then [y, z]δ(w(p)) = 0 for all y, z, p ∈ R .

Journal of Mathematical Extension, 2016

The main purpose of this article is to prove the following main result: Let R be a 2-torsion free semiprime ring and T : R → R be a Jordan left centralizer associated with an l-semi Hochschild 2-cocycle α: R ⨯ R → R. Then, T is a left centralizer associated with α. In order to show application of this result, several corollaries concerning Jordan generalized derivations, Jordan σ-derivations, Jordan generalized σ-derivations and Jordan (σ, τ )-derivations will be presented.

In this paper we establish the following result: "If R is a semiprime ring admitting a derivation d such that either (İ) xyx+d(xyx)=x* y + dix*y) or (ii) yxy + d (xyx)=xy s i.d (xy a ) for allx, yeR, then ii must be commutative." Further, if R is prime, then (i) or (ii) need only be assumed for all x, y in some non-zero ideal of R.

Turk J Math, 2010

We investigate some properties of generalized (α, β)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (α, β)-derivation, with associated (α, β)-derivation δ, on a semiprime ring R such that [g(x),α(x)] = 0 for all x ∈ R, then δ(x)[y, ...

Hacettepe Journal of Mathematics and Statistics

Let R be a ring with center Z and α, β and d mappings of R. A mapping F of R is called a centrally-extended multiplicative (generalized)-(α, β)-derivation associated with d if F (xy) − F (x)α(y) − β(x)d(y) ∈ Z for all x, y ∈ R. The objective of the present paper is to study the following conditions: (i) F (xy) ± β(x)G(y) ∈ Z, (ii) F (xy) ± g(x)α(y) ∈ Z and (iii) F (xy) ± g(y)α(x) ∈ Z for all x, y in some appropriate subsets of R, where G is a multiplicative (generalized)-(α, β)-derivation of R associated with the map g on R.

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2015

Let R be a semiprime ring and α any mapping on R. A mapping F : R → R is called multiplicative (generalized)-derivation if F(x y) = F(x)y + xd(y) for all x, y ∈ R, where d : R → R is any map (not necessarily additive). In this paper our main motive is to study the commutativity of semiprime rings and nature of mappings.

Aequationes mathematicae, 2005

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let R be a semiprime ring with center Z(R), and let f , g be derivations of R such that f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central.

2012

Let R be a ring with centre Z(R). An additive mapping F : R −→ R is said to be a generalized derivation if there exists a derivation d : R −→ R such that F(xy )= F(x)y+xd(y), for all x,y ∈ R (the map d is called the derivation associated with F). In the present note we prove that if a semiprime ring R admits a generalized derivation F, d is the nonzero associated derivation of F, satisfying certain polynomial constraints on a nonzero ideal I, then R contains a nonzero central ideal. Mathematics Subject Classification: Primary 16N60; Secondary 16W25