scholarly journals On semiderivations of *-prime rings

2014 ◽  
Vol 33 (2) ◽  
pp. 179-186
Author(s):  
Öznur Gölbaşı ◽  
Onur Ağırtıcı
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings

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.

scholarly journals Semi derivations of prime gamma rings

2012 ◽  
Vol 31 ◽  
pp. 65-70
Author(s):  
Kalyan Kumar Dey ◽  
Akhil Chandra Paul
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings ◽  
Subject Classifications ◽  
Associated Function ◽  
Gamma Rings

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

Some results on ∗-differential identities in prime rings

2021 ◽  
Author(s):  
Deepak Kumar ◽  
Bharat Bhushan ◽  
Gurninder S. Sandhu
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
Derivation Formula

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

scholarly journals Jordan Derivations on Lie Ideals of ?-Prime Rings

2017 ◽  
Vol 36 ◽  
pp. 1-5
Author(s):  
Akhil Chandra Paul ◽  
Md Mizanor Rahman
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Lie Ideal ◽  
Prime Rings ◽  
R And D ◽  
Square Closed Lie Ideal ◽  
Torsion Free

In this paper we prove that, if U is a s-square closed Lie ideal of a 2-torsion free s-prime ring R and  d: R(R is an additive mapping satisfying d(u2)=d(u)u+ud(u) for all u?U then d(uv)=d(u)v+ud(v) holds for all  u,v?UGANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 1-5

scholarly journals Semiderivations Satisfying Certain Algebraic Identities on Jordan Ideals

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Abdellah Mamouni ◽  
Lahcen Oukhtite
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings ◽  
Torsion Free

Let be a ring. An additive mapping is called semiderivation of if there exists an endomorphism of such that and , for all in . Here we prove that if is a 2-torsion free -prime ring and a nonzero -Jordan ideal of such that for all , then either is commutative or for all . Moreover, we initiate the study of generalized semiderivations in prime rings.

scholarly journals On Ideals and Commutativity of Prime Rings with Generalized Derivations

2018 ◽  
Vol 11 (1) ◽  
pp. 79 ◽  
Author(s):  
Mohammad Khalil Abu Nawas ◽  
Radwan M. Al-Omary
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities

An additive mapping F: R → R is called a generalized derivation on R if there exists a derivation d: R → R such that F(xy) = xF(y) + d(x)y holds for all x,y ∈ R. It is called a generalized (α,β)−derivation on R if there exists an (α,β)−derivation d: R → R such that the equation F(xy) = F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisfies certain differential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.

On Generalized (α,β)-Derivations in Prime Rings

2010 ◽  
Vol 17 (spec01) ◽  
pp. 865-874 ◽  
Author(s):  
Hidetoshi Marubayashi ◽  
Mohammad Ashraf ◽  
Nadeem-ur Rehman ◽  
Shakir Ali
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings

Let R be a ring and α,β be endomorphisms 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. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (α,β)-derivation F satisfying any one of the properties: (i) [F(x),x]α,β=0, (ii) F([x,y])=0, (iii) F(x ◦ y)=0, (iv) F([x,y])=[x,y]α,β, (v) F(x ◦ y)=(x ◦ y)α,β, (vi) F(xy)- α(xy) ∈ Z(R), (vii) F(x)F(y)- α(xy) ∈ Z(R) for all x, y in an appropriate subset of R.

scholarly journals On Jordan ideals and left(θ,θ)-derivations in prime rings

2004 ◽  
Vol 2004 (37) ◽  
pp. 1957-1964 ◽  
Author(s):  
S. M. A. Zaidi ◽  
Mohammad Ashraf ◽  
Shakir Ali
Keyword(s):  
Prime Ring ◽  
Nonempty Subset ◽  
Additive Mapping ◽  
Prime Rings ◽  
Torsion Free

LetRbe a ring andSa nonempty subset ofR. Suppose thatθandϕare endomorphisms ofR. An additive mappingδ:R→Ris called a left(θ,ϕ)-derivation (resp., Jordan left(θ,ϕ)-derivation) onSifδ(xy)=θ(x)δ(y)+ϕ(y)δ(x)(resp.,δ(x2)=θ(x)δ(x)+ϕ(x)δ(x)) holds for allx,y∈S. Suppose thatJis a Jordan ideal and a subring of a2-torsion-free prime ringR. In the present paper, it is shown that ifθis an automorphism ofRsuch thatδ(x2)=2θ(x)δ(x)holds for allx∈J, then eitherJ⫅Z(R)orδ(J)=(0). Further, a study of left(θ,θ)-derivations of a prime ringRhas been made which acts either as a homomorphism or as an antihomomorphism of the ringR.

Generalized Jordan Semiderivations in Prime Rings

2015 ◽  
Vol 58 (2) ◽  
pp. 263-270 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Abdellah Mamouni ◽  
Lahcen Oukhtite
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings ◽  
R And D

AbstractLet R be a ring and let g be an endomorphism of R. The additive mapping d: R → R is called a Jordan semiderivation of R, associated with g, ifd(x2) = d(x)x + g(x)d(x) = d(x)g(x) + xd(x) and d(g(x)) = g(d(x))for all x ∊ R. The additive mapping F: R → R is called a generalized Jordan semiderivation of R, related to the Jordan semiderivation d and endomorphism g, ifF(x2) = F(x)x + g(x)d(x) = F(x)g(x) + xd(x) and F(g(x)) = g(F(x))for all x ∊ R. In this paper we prove that if R is a prime ring of characteristic different from 2, g an endomorphism of R, d a Jordan semiderivation associated with g, F a generalized Jordan semiderivation associated with d and g, then F is a generalized semiderivation of R and d is a semiderivation of R. Moreover, if R is commutative, then F = d.

scholarly journals On (?,?)-Derivations and Commutativity of Prime and Semi prime ?-rings

2016 ◽  
Vol 13 (1) ◽  
pp. 198-203
Author(s):  
Baghdad Science Journal
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings ◽  
R And D ◽  
Strong Commutativity Preserving ◽  
Strong Commutativity

Let R be a ?-ring, and ?, ? be two automorphisms of R. An additive mapping d from a ?-ring R into itself is called a (?,?)-derivation on R if d(a?b) = d(a)? ?(b) + ?(a)?d(b), holds for all a,b ?R and ???. d is called strong commutativity preserving (SCP) on R if [d(a), d(b)]? = [a,b]_?^((?,?) ) holds for all a,b?R and ???. In this paper, we investigate the commutativity of R by the strong commutativity preserving (?,?)-derivation d satisfied some properties, when R is prime and semi prime ?-ring.

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

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