ArticlePDF Available

Ideals and Symmetrc Left Bi-Derivations on Prime Rings

Authors:
C.Jaya Subba Reddy at Sri Venkateswara University
C.Jaya Subba Reddy
G. Venkata Bhaskara Rao
G. Venkata Bhaskara Rao
G. Venkata Bhaskara Rao
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Download full-text PDF
Read full-text
Download citation
Content uploaded by C.Jaya Subba Reddy
Author content
All content in this area was uploaded by C.Jaya Subba Reddy on May 30, 2022
Content may be subject to copyright.
Research J. Science and Tech. 9(4): October-December, 2017
601
ISSN 0975-4393 (PRINT)
2349-2988 (ONLINE)
DOI: 10.5958/2349-2988.2017.00102.4
Vol. 09| Issue-04|
October -December | 2017
Available online at
www.anvpublication.org
Research Journal of Science and Technology
Home page www.rjstonline.com
RESEARCH ARTICLE
Ideals and Symmetrc Left Bi-Derivations on Prime Rings
Dr. C. Jaya Subba Reddy, G. Venkata Bhaskara Rao
Department of Mathematics, Sri Venkateswara University, Tirupati-517502, Andhra Pradesh, India.
*CorrespondingAuthorE-mail:cjsreddysvu@gmail.com
ABSTRACT:
Let be a non commutative 2, 3-torsion free prime ring and be a non zero ideal of . Let .,.:×  
be a symmetric left bi-derivation such that (,)  and is a trace of . If (i),= 0, for all   , (ii)
, (), for all   , then = 0. Suppose that there exists symmetric left bi-derivations 1.,.:×
   and 2.,.:×   and .,.:×   is a symmetric bi-additive mapping, such that (i)
12,= 0, for all  , (ii) 12=(), for all  , where 1 and 2 are the traces of 1 and
2 respectively and is trace of , then either 1= 0 or 2= 0. If acts as a left (resp. right) -
homomorphism on , then = 0.
KEYWORDS:
Prime ring, Symmetric mapping, Trace, Bi-additive mapping, Symmetric bi-additive mapping, Symmetric bi-
derivation, Symmetric left bi-derivation.
INTRODUCTION:
The concept of a symmetric bi-derivation has been
introduced by Maksa.Gy in [5],[6]. A classical result in
the theory of centralizing mappings is a theorem first
proved by Posner.E [7]. Vukman.J [8] has studied some
results concerning symmetric bi-derivations on prime
and semi prime rings. Yenigul.M.S and Argac.N [9]
proved that the results which are obtained in [8,
Theorems 1, 2, 3] by using a nonzero ideal of . Jaya
SubbaReddy.C [2], [3] has proved some results
concerning symmetric left bi-derivation on prime rings.
In this paper we proved some results concerning to
ideals and symmetric left bi-derivations on prime rings.
Throughout this paper will be associative. We shall
denote by () the center of a ring . Recall that a ring
is prime if  = (0) implies that = 0or = 0.
We shall write [,] for   and use the
identities,=,+[,], ,=,+
[,]. An additive map :   is called derivation if
=+(), for all,  .
Received on 21.09.2017 Modified on 06.11.2017
Accepted on 05.12.2017 ©A&V Publications All right reserved
Research J. Science and Tech. 2017; 9(4): 601-604.
DOI: 10.5958/2349-2988.2017.00102.4
A mapping .,.:×   is said to be symmetric if
,= (,), for all,  . A mapping :  
defined by =(,), where .,.:×   is
a symmetric mapping, is called a trace of B. It is obvious
that, in case .,.:×   is symmetric mapping
which is also bi-additive (i. e. additive in both
arguments) the trace of satisfies the relation +
=++ 2(,), for all,  .We shall
use also the fact that the trace of a symmetric bi-additive
mapping is an even function. A symmetric bi-additive
mapping .,.:×   is called a symmetric bi-
derivation if ,=,+(,), for all
,,  . Obviously, in this case also the relation
,=,+(,), for all,,  . A
symmetric bi-additive mapping .,.:×   is
called a symmetric left bi-derivation if ,=
,+,, for all ,,  . Obviously, in
this case also the relation ,=,+
,, for all ,,  . A mapping:   is said
to be commuting on if ,= 0, for all  . A
mapping :   is said to be centralizing on if
, (), for all  . A ring is said to be n-
torsion free if whenever  = 0, with   , then =
0, where is nonzero integer. Let be a ring and be a
non zero left (right) ideal of .
... And , if ( ∝ , ) = ( , ) ∝ + ∝ ( , ) for all ∈ and ∝∈ Γ , then d is called a Jordan bi-derivation on M×M into M . The notion of symmetric bi-derivation was introduced by G.Maksa [8] and [5] An bi-additive map : × → is said to be a generalized symmetric bi-derivation on M×M into M if there exists symmetric bi-derivation : × → such that ( ∝ , ) = ( , ) ∝ + ∝ ( , ) is satisfied for all , , ∈ and ∝∈ Γ . And , F is said to be a Jordan generalized bi-derivation on M×M into M if there exists a Jordan biderivation : × → such that ( ∝ , ) = ( , ) ∝ + ∝ ( , ) holds for all , ∈ and ∝∈ Γ . ...
ON JORDAN GENERALIZED HIGHER BI-DERIVATIONS ON PRIME GAMMA RINGS
Research
Full-text available
  • Dec 2019
In this study , we define the concepts of a generalized higher bi-derivation , Jordan generalized higher bi-derivation and Jordan triple generalized higher bi-derivation on Г-rings and show that a Jordan generalized higher bi-derivation on 2-torsion free prime Г-ring is a generalized higher bi-derivation. 1.Introduction Let M and Г be two additive abelian groups. If there exists a mapping (í µí±Ž, ∝, í µí±) → í µí±Ž ∝ í µí± of í µí±€ × Г × í µí±€ → í µí±€ satisfying the following for all a, b, c ∈ í µí±€ and ∝, í µí»½ ∈ Г ∶ (í µí±–) (í µí±Ž + í µí±) ∝ í µí± = í µí±Ž ∝ í µí± + í µí± ∝ í µí± , í µí±Ž (∝ +í µí»½)í µí± = í µí±Ž ∝ í µí± + í µí±Ží µí»½í µí± , í µí±Ž ∝ (í µí± + í µí±) = í µí±Ž ∝ í µí± + í µí±Ž ∝ í µí± and (í µí±–í µí±–) (í µí±Ž ∝ í µí±)í µí»½í µí± = í µí±Ž ∝ (í µí±í µí»½í µí±). Then M is called a Г − í µí±Ÿí µí±–í µí±›í µí±” The notion of a Г − í µí±Ÿí µí±–í µí±›í µí±” was introduced by Nobusawa [9] and generalized by Barnes [2] as defined above. Many properties of Г − í µí±Ÿí µí±–í µí±›í µí±” were obtained by Barnes [2] , kyuno [6] , Luh [7] and others. let M be a Г − í µí±Ÿí µí±–í µí±›í µí±”. then M is called 2-torsion free if 2a= 0 implies a= 0 for all í µí±Ž ∈ í µí±€. Besides , M is called a prime Г − í µí±Ÿí µí±–í µí±›í µí±” if , for all í µí±Ž, í µí± ∈ í µí±€, í µí±Ž Г í µí±€ Г í µí± = (0) implies either a= 0 or b= 0. and, M is called semiprime if í µí±Ž Г í µí±€ Г í µí±Ž = (0) with í µí±Ž ∈ í µí±€ implies a= 0. Note that every prime Г − í µí±Ÿí µí±–í µí±›í µí±” is obviously semiprime. M is said to be a commutative Γ − í µí±Ÿí µí±–í µí±›í µí±” if í µí±Ž ∝ í µí± = í µí± ∝ í µí±Ž holds for all í µí±Ž, í µí± ∈ í µí±€ and ∝∈ Γ. Let M be a Γ − í µí±Ÿí µí±–í µí±›í µí±”. then , for í µí±Ž, í µí± ∈ í µí±€ and ∝∈ Γ , we define [í µí±Ž, í µí±] ∝ = í µí±Ž ∝ í µí± − í µí± ∝ í µí±Ž , known as the commutator of í µí±Ž í µí±Ží µí±›í µí±‘ í µí± with respect to ∝. The notion of derivation and Jordan derivation on a Γ-ring were defined by M. Sapanci and A. Nakajima in [11], as follow An additive mapping í µí±‘: í µí±€ → í µí±€ is called a derivation of M if í µí±‘(í µí±Ž ∝ í µí±) = í µí±‘(í µí±Ž) ∝ í µí± + í µí±Ž ∝ í µí±‘(í µí±) for all í µí±Ž, í µí± ∈ í µí±€ , ∝∈ Γ. And , if í µí±‘(í µí±Ž ∝ í µí±Ž) = í µí±‘(í µí±Ž) ∝ í µí±Ž + í µí±Ž ∝ í µí±‘(í µí±Ž) for all í µí±Ž ∈ í µí±€ and ∝∈ Γ , then d is called a Jordan derivation of M. The concept of Jordan generalized derivation of a Γ-ring has been developed by Y.Ceven and M.A.Ozturk in [3] ,as follow An additive map í µí°¹: í µí±€ → í µí±€ is said to be a generalized derivation of M if there exists a derivation í µí±‘: í µí±€ → í µí±€ such that í µí°¹(í µí±Ž ∝ í µí±) = í µí°¹(í µí±Ž) ∝ í µí± + í µí±Ž ∝ í µí±‘(í µí±) is satisfied for all í µí±Ž, í µí± ∈ í µí±€ and ∝∈ Γ. And , F is said to be a Jordan generalized derivation of M if there
View
View
ResearchGate has not been able to resolve any references for this publication.