Results on multiplicative semiderivations in semiprime rings (original) (raw)

Some results On Semiderivations of Semiprime Semirings

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.

Some theorems of commutativity on semiprime rings with mappings

2016

Let R be a semiprime ring and F : R → R a mapping such that F (xy) = F (y)x + yd(x) for all x, y ∈ R, where d is any map on R. In this paper, we investigate the commutativity of semiprime rings with a mapping F on R. Several theorems of commutativity of semiprime rings are obtained.

A Commutativity theorem for semiprime rings

Journal of the Australian Mathematical Society, 1980

It is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y e R, there exists a positive integer n depending on v and y such that (\_v)*-x*>'*is central for k = n,n+ 1,H + 2, then R is commutative, thus generalizing a result of Kaya.

On symmetric biadditive mappings of semiprime rings

Boletim da Sociedade Paranaense de Matemática, 2015

Let R be a ring with centre Z(R). A mapping D(., .) : R× R −→ R issaid to be symmetric if D(x, y) = D(y, x) for all x, y ∈ R. A mapping f : R −→ Rdefined by f(x) = D(x, x) for all x ∈ R, is called trace of D. It is obvious thatin the case D(., .) : R × R −→ R is a symmetric mapping, which is also biadditive(i.e. additive in both arguments), the trace f of D satisfies the relation f(x + y) =f(x) + f(y) + 2D(x, y), for all x, y ∈ R. In this paper we prove that a nonzero left idealL of a 2-torsion free semiprime ring R is central if it satisfies any one of the followingproperties: (i) f(xy) ∓ [x, y] ∈ Z(R), (ii) f(xy) ∓ [y, x] ∈ Z(R), (iii) f(xy) ∓ xy ∈Z(R), (iv) f(xy)∓yx ∈ Z(R), (v) f([x, y])∓[x, y] ∈ Z(R), (vi) f([x, y])∓[y, x] ∈ Z(R),(vii) f([x, y])∓xy ∈ Z(R), (viii) f([x, y])∓yx ∈ Z(R), (ix) f(xy)∓f(x)∓[x, y] ∈ Z(R),(x) f(xy)∓f(y)∓[x, y] ∈ Z(R), (xi) f([x, y])∓f(x)∓[x, y] ∈ Z(R), (xii) f([x, y])∓f(y)∓[x, y] ∈ Z(R), (xiii) f([x, y])∓f(xy)∓[x, y] ∈ Z(R), (xiv) f([x, y])∓f(xy)∓[y, x] ...

A General Characterization of Additive Maps on Semiprime Rings

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.

On semiderivations of *-prime rings

Boletim da Sociedade Paranaense de Matemática, 2014

Let R be a ∗-prime ring with involution ∗ and center Z(R). An additive mapping F:R→R is called a semiderivation if there exists a function g:R→R such that (i) F(xy)=F(x)g(y)+xF(y)=F(x)y+g(x)F(y) and (ii) F(g(x))=g(F(x)) hold for all x,y∈R. In the present paper, some well known results concerning derivations of prime rings are extended to semiderivations of ∗-prime rings.

Multiplicative (generalized)-derivation in semiprime rings

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.