scholarly journals Derivations on Lie ideals of completely semiprime ? -rings

2016 ◽  
Vol 27 (1) ◽  
pp. 51-61
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Semiprime Ring ◽  
Jordan Derivation ◽  
Lie Ideal ◽  
Semiprime Rings ◽  
Torsion Free

The aim of the paper is to prove the following theorem concerning a class of ? -rings. LetM be a 2-torsion free completely semiprime ? -ring satisfying the condition a?b?c = a?b?c, for all a,b,c?M and ? ,? ?? ; U be an admissible Lie ideal ofM . If d :M ?M is a Jordan derivation on U of M , then d is a derivation on U of M .Bangladesh J. Sci. Res. 27(1): 51-61, June-2014

scholarly journals Jordan derivations on 2-Torsion free semiprime ?-Rings

2014 ◽  
Vol 38 (2) ◽  
pp. 189-195
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Semiprime Ring ◽  
Jordan Derivation ◽  
Semiprime Rings ◽  
Academy Of Sciences ◽  
Torsion Free

The objective of this paper was to study Jordan derivations on semiprime ?-ring. Let M be a 2-torsion free semiprime ?-ring satisfying the condition a?b?c = a?b?c for all a,b,c ? M and ?, ? ? ?. The authors proved that every Jordan derivation of M is a derivation of M. DOI: http://dx.doi.org/10.3329/jbas.v38i2.21343 Journal of Bangladesh Academy of Sciences, Vol. 38, No. 2, 189-195, 2014

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

2016 ◽  
Vol 34 ◽  
pp. 35-46
Author(s):  
Md Mizanor Rahman ◽  
Akhil Chandra Paul
Keyword(s):  
Additive Mapping ◽  
Jordan Derivation ◽  
Lie Ideal ◽  
Semiprime Rings ◽  
Torsion Free

Let M be a 2-torsion free semiprime G-ring satisfying the condition a?b?c = a?b?c,?a, b, c ?M and ?, ? ??. Let U be an admissible Lie ideal of M that is, u?u ? U,?u ? U, ? ?G and U ?Z(M), the centre of M. If d : M -> M is an additive mapping such that d is a Jordan derivation on U of M, then d is a derivation on U.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 35-46

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

A NOTE ON -JORDAN DERIVATIONS OF RINGS AND BANACH ALGEBRAS

2015 ◽  
Vol 93 (2) ◽  
pp. 231-237 ◽  
Author(s):  
IRENA KOSI-ULBL ◽  
JOSO VUKMAN
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Torsion Free

In this paper we prove the following result: let$m,n\geq 1$be distinct integers, let$R$be an$mn(m+n)|m-n|$-torsion free semiprime ring and let$D:R\rightarrow R$be an$(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation$(m+n)D(x^{2})=2mD(x)x+2nxD(x)$for$x\in R$. Then$D$is a derivation which maps$R$into its centre.

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 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 Jordan Derivations on Completely Semiprime Gamma–Rings

2016 ◽  
Vol 34 ◽  
pp. 21-26
Author(s):  
Md Mizanor Rahman ◽  
Akhil Chandra Paul
Keyword(s):  
Semiprime Ring ◽  
Suitable Condition ◽  
Jordan Derivation ◽  
Gamma Rings ◽  
Torsion Free

In this paper we prove that under a suitable condition every Jordan derivation on a 2-torsion free completely semiprime ?-ring is a derivation.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 21-26

scholarly journals (U. M)-derivations in completely semiprime ?-rings

2016 ◽  
Vol 51 (1) ◽  
pp. 69-74
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Semiprime Ring ◽  
Subject Classification ◽  
Lie Ideal ◽  
Semiprime Rings ◽  
And Mathematics

The objective of this paper is to extend and generalize some results of (Rahman and Paul, 2014) in completely semiprime ?- rings. We prove that, if is an admissible Lie ideal of a completely semiprime ?-ring and d is a (U, M) -derivation of then d(u?v)=d(u)?v+u?d(v) for all u, v ? U and ? ? ?Mathematics Subject Classification: 13N15, 16W10, 17C50Bangladesh J. Sci. Ind. Res. 51(1). 69-74, 2016

scholarly journals On symmetric biadditive mappings of semiprime rings

2017 ◽  
Vol 35 (1) ◽  
pp. 9
Author(s):  
Asma Ali ◽  
Khalid Ali Hamdin ◽  
Shahoor Khan
Keyword(s):  
Semiprime Ring ◽  
Semiprime Rings ◽  
Torsion Free

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] ∈ Z(R),(xv) f(x)f(y) ∓ [x, y] ∈ Z(R), (xvi) f(x)f(y) ∓ [y, x] ∈ Z(R), (xvii) f(x)f(y) ∓ xy ∈Z(R), (xviii) f(x)f(y) ∓ yx ∈ Z(R), (xix) f(x) ◦ f(y) ∓ [x, y] ∈ Z(R), (xx) f(x) ◦f(y) ∓ xy ∈ Z(R), (xxi) f(x) ◦ f(y) ∓ yx ∈ Z(R), (xxii) f(x)f(y) ∓ x ◦ y ∈ Z(R),(xxiii) [x, y] − f(xy) + f(yx) ∈ Z(R), for all x, y ∈ R, where f stands for the trace of asymmetric biadditive mapping D(., .) : R × R −→ R.

On generalized (θ, φ)-derivations in semiprime rings with involution

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Mohammad Ashraf ◽  
Nadeem-ur-Rehman ◽  
Shakir Ali ◽  
Muzibur Mozumder
Keyword(s):  
Semiprime Ring ◽  
Additive Mapping ◽  
Semiprime Rings ◽  
Rings With Involution ◽  
Torsion Free

AbstractThe main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained.

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