scholarly journals Generalized Derivations and Left Ideals in Prime and Semiprime Rings

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Basudeb Dhara ◽  
Atanu Pattanayak
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

Let R be an associative ring, λ a nonzero left ideal of R, d:R→R a derivation and G:R→R a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) G(x∘y)=a(xy±yx); (2) G[x,y]=a(xy±yx); (3) d(x)∘d(y)=a(xy±yx); for all x,y∈λ and a∈{0,1,-1}.

scholarly journals Generalized (; )-derivations and Left Ideals in Prime and Semiprime Rings

2017 ◽  
Vol 13 (2) ◽  
pp. 7163-7167
Author(s):  
Asma Ali ◽  
Hamidur Rahaman
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

Let R be an associative ring, ; be the automorphisms of R, be a nonzero left ideal of R, F : R ! R be a generalized (; )-derivation and d : R ! Rbe an (; )-derivation. In the present paper we discuss the following situations: (i) F(xoy) = a(xy yx), (ii) F([x; y]) = a(xy yx), (iii) d(x)od(y) = a(xy yx) forall x; y 2 and a 2 f0; 1;ô€€€1g. Also some related results have been obtained.

ON n-CENTRALIZING GENERALIZED DERIVATIONS IN SEMIPRIME RINGS WITH APPLICATIONS TO C*-ALGEBRAS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250111 ◽  
Author(s):  
BASUDEB DHARA ◽  
SHAKIR ALI
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
C Algebras ◽  
Semiprime Rings ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xn – xn f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

scholarly journals Remarks on Generalized Derivations in Prime and Semiprime Rings

2010 ◽  
Vol 2010 ◽  
pp. 1-6 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

LetRbe a ring with centerZandIa nonzero ideal ofR. An additive mappingF:R→Ris called a generalized derivation ofRif there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)for allx,y∈R. In the present paper, we prove that ifF([x,y])=±[x,y]for allx,y∈IorF(x∘y)=±(x∘y)for allx,y∈I, then the semiprime ringRmust contains a nonzero central ideal, providedd(I)≠0. In caseRis prime ring,Rmust be commutative, providedd≠0. The cases (i)F([x,y])±[x,y]∈Zand (ii)F(x∘y)±(x∘y)∈Zfor allx,y∈Iare also studied.

scholarly journals Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Nadeem UR Rehman ◽  
Abu Zaid Ansari
Keyword(s):  
Lie Ideal ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers ◽  
Torsion Free

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.

scholarly journals Some identities involving multiplicative generalized derivations in orime and semiprime rings

2018 ◽  
Vol 36 (1) ◽  
pp. 25 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Generalized Derivation ◽  
Generalized Derivations ◽  
Semiprime Rings

Let $R$ be a ring with center $Z(R)$. A mapping $F:R\rightarrow R$ is called a multiplicative generalized derivation, if $F(xy)=F(x)y+xg(y)$ is fulfilled for all $x,y\in R$, where $g:R\rightarrow R$ is a derivation. In the present paper, our main object is to study the situations: (1) $F(xy)- F(x)F(y)\in Z(R)$, (2) $F(xy)+ F(x)F(y)\in Z(R)$, (3) $F(xy)- F(y)F(x)\in Z(R)$, (4) $F(xy)+ F(y)F(x)\in Z(R)$, (5) $F(xy)- g(y)F(x)\in Z(R)$; for all $x,y$ in some suitable subset of $R$.

scholarly journals On Generalized ()-Derivations in Semiprime Rings

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Basudeb Dhara ◽  
Atanu Pattanayak
Keyword(s):  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Semiprime Rings

Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .

Centralizing and Commuting Left Generalized Derivations on Prime Rings

2015 ◽  
Vol 11 ◽  
pp. 1-3 ◽  
Author(s):  
C. Jaya Subba Reddy ◽  
S. Mallikarjuna Rao ◽  
V. Vijaya Kumar
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime Rings

Let R be a prime ring and d a derivation on R. If is a left generalized derivation on R such that ƒ is centralizing on a left ideal U of R, then R is commutative.

Generalized derivations preserving Engel condition in prime and semiprime rings

2020 ◽  
pp. 2150041
Author(s):  
Rita Prestigiacomo
Keyword(s):  
Prime Ring ◽  
Algebraic Closure ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Martindale Quotient Ring ◽  
Engel Condition

Let [Formula: see text] be a prime ring with [Formula: see text], [Formula: see text] a non-central Lie ideal of [Formula: see text], [Formula: see text] its Martindale quotient ring and [Formula: see text] its extended centroid. Let [Formula: see text] and [Formula: see text] be nonzero generalized derivations on [Formula: see text] such that [Formula: see text] Then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text], for any [Formula: see text], unless [Formula: see text], where [Formula: see text] is the algebraic closure of [Formula: see text].

scholarly journals On Maps of Period 2 on Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
H. E. Bell ◽  
M. N. Daif
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Period 2 ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Rings With Involution

A mapfof the ringRinto itself is of period 2 iff2x=xfor allx∈R; involutions are much studied examples. We present some commutativity results for semiprime and prime rings with involution, and we study the existence of derivations and generalized derivations of period 2 on prime and semiprime rings.

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