Some results On Semiderivations of Semiprime Semirings

International Journal of Algebra

Let R be a semiprime ring. An additive mapping d : R → R is called a semiderivation if there exists a function g : R → R such that (i) d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and (ii) d(g(x)) = g(d(x)) hold for all x, y ∈ R. The aim of this paper is to explore the commutativity of semiprime rings admitting multiplicative semiderivations.

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.

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.

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

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.

In this paper, we study the class of Right regular and Multiplicatively subidempotent semirings. Especially we have focused on the additive identity ‘e’ which is also multiplicative identity in both semirings.

Let M be a prime Γ-ring satisfying a certain assumption (*). An additive mapping f : M → M is a semi-derivation if f(xαy) = f(x)αg(y) + xαf(y) = f(x)αy + g(x)αf(y) and f(g(x)) = g(f(x)) for all x, y∈M and α∈Γ, where g : M→M is an associated function. In this paper, we generalize some properties of prime rings with semi-derivations to the prime Γ-rings with semi-derivations. 2000 AMS Subject Classifications: 16A70, 16A72, 16A10. 1. Introduction J. C. Chang [6] worked on semi-derivations of prime rings. He obtained some results of derivations of prime rings into semi-derivations. H. E. Bell and W. S. Martindale III [1] investigated the commutativity property of a prime ring by means of semi-derivations. C. L. Chuang [7] studied on the structure of semi-derivations in prime rings. He obtained some remarkable results in connection with the semi-derivations. J. Bergen and P. Grzesczuk [3] obtained the commutativity properties of semiprime rings with the help of skew (semi)-derivations. A...

2018

Motivated by some results on Reverse, Jordan and Left Biderivations , in [5], the authors investigated a prime ring of characteristic 2,3 that admits a nonzero Jordan left biderivation is commutative. Also, if R is a semiprime ring and B is a left biderivation, then B must be an ordinary biderivation that maps R into its center. In this paper, we also derived the same thing in semiprime semirings.

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.

2021

Let R be a prime ring with the extended centroid C and the Matrindale quotient ring Q. An additive mapping F : R → R is called a semiderivation associated with a mapping G : R → R, whenever F (xy) = F (x)G (y) + xF (y) = F (x)y + G (x)F (y) and F (G (x)) = G (F (x)) holds for all x, y ∈ R. In this manuscript, we investigate and describe the structure of a prime ring R which satisfies F (x ◦ y) ∈ Z (R) for all x, y ∈ R, where m,n ∈ Z and F : R → R is a semiderivation with an automorphism ξ of R. Further, as an application of our ring theoretic results, we discussed the nature of C ∗-algebras. To be more specific, we obtain for any primitive C ∗-algebra A . If an anti-automorphism ζ : A → A satisfies the relation (x) +x ∈ Z (A ) for every x, y ∈ A , then A is C ∗ −W4-algebra, i. e., A satisfies the standard identity W4(a1, a2, a3, a4) = 0 for all a1, a2, a3, a4 ∈ A .