On semiderivations in prime gamma rings

GANIT: Journal of Bangladesh Mathematical Society, 2012

Let M be a prime ?-ring satisfying a certain assumption (*). An additive mapping f : M ? M is a semi-derivation if f(x?y) = f(x)?g(y) + x?f(y) = f(x)?y + g(x)?f(y) and f(g(x)) = g(f(x)) for all x, y?M and ? ? ?, where g : M?M is an associated function. In this paper, we generalize some properties of prime rings with semi-derivations to the prime &Gamma-rings with semi-derivations. 2000 AMS Subject Classifications: 16A70, 16A72, 16A10.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10309GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 65-70

Boletim da Sociedade Paranaense de Matemática, 2010

Discussiones Mathematicae - General Algebra and Applications, 2015

In the present paper, it is introduced the definition of a reverse derivation on a Γ-ring M. It is shown that a mapping derivation on a semiprime Γ-ring M is central if and only if it is reverse derivation. Also it is shown that M is commutative if for all a, b ∈ I (I is an ideal of M) satisfying d(a) ∈ Z(M), and d(a • b) = 0.

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 .

Aequationes mathematicae

Let S be a semiprime semiring. An additive mapping is called a semi derivation if there exists a function such that (i) , (ii) hold for all . In this paper we try to generalize some properties of prime rings with derivations to semiprime semirings with semiderivations.

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, ...

Boletim da Sociedade Paranaense de Matemática, 2014

Let R be an associative ring, I a nonzero ideal of R and σ, τ two epimorphisms of R. An additive mapping F : R → R is called a generalized (σ, τ)derivation of R if there exists a (σ, τ)-derivation d : R → R such that F (xy) = F (x)σ(y) + τ (x)d(y) holds for all x, y ∈ R. The objective of the present paper is to study the following situations in prime and semiprime rings: (i) [F (x), x]σ,τ = 0, (ii) F ([x, y]) = 0, (iii) F (x • y) = 0, (iv) F ([x, y]) = [x, y]σ,τ , (v) F (x • y) = (x • y)σ,τ , (vi) F (xy) − σ(xy) ∈ Z(R), (vii) F (x)F (y) − σ(xy) ∈ Z(R) for all x, y ∈ I, when F is a generalized (σ, τ)-derivation of R.

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.

International Journal of Mathematics and Mathematical Sciences, 2004

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.