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 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 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.

scholarly journals On Higher N-Derivation Of Prime Rings

2014 ◽  
Vol 11 (2) ◽  
pp. 211-219
Author(s):  
Baghdad Science Journal
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Torsion Free

The main purpose of this work is to introduce the concept of higher N-derivation and study this concept into 2-torsion free prime ring we proved that:Let R be a prime ring of char. 2, U be a Jordan ideal of R and be a higher N-derivation of R, then , for all u U , r R , n N .

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 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 Derivations on Lie Ideals of σ-Prime Γ-Rings

2015 ◽  
Vol 39 (2) ◽  
pp. 249-255
Author(s):  
Md Mizanor Rahman ◽  
Akhil Chandra Paul
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Lie Ideal ◽  
Square Closed Lie Ideal ◽  
Academy Of Sciences ◽  
Gamma Rings ◽  
Torsion Free

The authors extend and generalize some results of previous workers to ?-prime ?-ring. For a ?-square closed Lie ideal U of a 2-torsion free ?-prime ?-ring M, let d: M ?M be an additive mapping satisfying d(u?u)=d(u)? u + u?d(u) for all u ? U and ? ? ?. The present authors proved that d(u?v) = d(u)?v + u?d(v) for all u, v ? U and ?? ?, and consequently, every Jordan derivation of a 2-torsion free ?-prime ?-ring M is a derivation of M.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 2, 249-255, 2015

scholarly journals On the commutativity of semi-prime rings

1982 ◽  
Vol 32 (1) ◽  
pp. 48-51
Author(s):  
Abdullah H. Al-Moajil
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Torsion Free

AbstractIt is shown that ifRis a 2-torsion-free semi-prime ring such that [xy, [xy, yx]] = 0 for allx, y∈R, thenRis commutative.

On Prime and Semiprime Rings with Derivations

2006 ◽  
Vol 13 (03) ◽  
pp. 371-380 ◽  
Author(s):  
Nurcan Argaç
Keyword(s):  
Prime Ring ◽  
Nonempty Subset ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

scholarly journals On Jordan ideals and Generalized (α, 1)- Reverse derivations in ∗-prime rings

2021 ◽  
Vol 23 (11) ◽  
pp. 236-242
Author(s):  
Sk. Haseena ◽  
◽  
Chennupalle Divya ◽  
C. Jaya Subba Reddy ◽  
◽  
...  
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Torsion Free

Let R will be a 2- torsion free ∗-prime ring and α be an automorphisum of R. F be a nonzero generalized (α, 1)- reverse derivation of R with associated nonzero (α, 1)- reverse derivation d which commutes with ∗ and J be a nonzero ∗-Jordan ideal and a subring of R. In the present paper, we shall prove that R is commutative if any one of the following holds: (i) [F(u), u]α,1 = 0, (ii) F(u) α(u) = ud(u), (iii) F(u2) = ± α(u2), (iv) F(u2) = 2d(u) α(u), (v) d(u2) = 2F(u) α(u), for all u ∈ U.

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