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

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 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 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 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

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.

Commutativity Conditions on Rings with Involution

1982 ◽  
Vol 34 (1) ◽  
pp. 17-22
Author(s):  
Paola Misso
Keyword(s):  
Semiprime Ring ◽  
Central Elements ◽  
Rings With Involution ◽  
Image Position ◽  
Partial Extension ◽  
Torsion Free

Let R be a ring with involution *. We denote by S, K and Z = Z(R) the symmetric, the skew and the central elements of R respectively.In [4] Herstein defined the hypercenter T(R) of a ring R asand he proved that in case R is without non-zero nil ideals then T(R) = Z(R).In this paper we offer a partial extension of this result to rings with involution.We focus our attention on the following subring of R:(We shall write H(R) as H whenever there is no confusion as to the ring in question.)Clearly H contains the central elements of R. Our aim is to show that in a semiprime ring R with involution which is 2 and 3-torsion free, the symmetric elements of H are central.

scholarly journals Generalized Derivations in Semiprime Gamma Rings

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Kalyan Kumar Dey ◽  
Akhil Chandra Paul ◽  
Isamiddin S. Rakhimov
Keyword(s):  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Gamma Rings ◽  
Torsion Free

LetMbe a 2-torsion-free semiprimeΓ-ring satisfying the conditionaαbβc=aβbαcfor alla,b,c∈M,  α,β∈Γ, and letD:M→Mbe an additive mapping such thatD(xαx)=D(x)αx+xαd(x)for allx∈M,  α∈Γand for some derivationdofM. We prove thatDis a generalized derivation.

scholarly journals On a Functional Equation Related to Two-Sided Centralizers

2018 ◽  
Vol 32 (1) ◽  
pp. 227-235
Author(s):  
Irena Kosi-Ulbl
Keyword(s):  
Functional Equation ◽  
Semiprime Ring ◽  
Additive Mapping ◽  
Fixed Integer ◽  
Torsion Free

Abstract The main aim of this manuscript is to prove the following result. Let n > 2 be a fixed integer and R be a k-torsion free semiprime ring with identity, where k ∈ {2, n - 1, n}. Let us assume that for the additive mapping T : R→ R 3T(xn) = T(x)xn-1+ xT(xn-2)x + xn-1T(x), x ∈R, is also fulfilled. Then T is a two-sided centralizer.

scholarly journals On Centralizers of 2-torsion Free Semiprime Gamma Rings

2021 ◽  
pp. 2351-2356
Author(s):  
Abdulkareem T. Mutlak ◽  
Abulrahman H. Majeed
Keyword(s):  
Additive Mapping ◽  
Left And Right ◽  
Gamma Rings ◽  
Torsion Free

In this paper, we prove that; Let M be a 2-torsion free semiprime  which satisfies the condition  for all  and α, β . Consider that  as an additive mapping such that  holds for all  and α , then T is a left and right centralizer.

scholarly journals Higher commutators, ideals and cardinality

1996 ◽  
Vol 54 (1) ◽  
pp. 41-54 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Finite Index ◽  
Associative Ring ◽  
Semiprime Ring ◽  
Torsion Free ◽  
The Ideal

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [[R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2cardT and there is a central ideal of R of index at most 2cardT in R.

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