Hindawi Publishing Corporation
ISRN Algebra
Volume 2013, Article ID 572690, 5 pages
http://dx.doi.org/10.1155/2013/572690
Research Article
Some Theorems for Sigma Prime Rings with
Differential Identities on Sigma Ideals
Mohd Rais Khan and Mohd Mueenul Hasnain
Department of Mathematics, Jamia Millia Islamia, Jamia Nagar, New Delhi 110025, India
Correspondence should be addressed to Mohd Mueenul Hasnain; mhamu786@gmail.com
Received 26 September 2013; Accepted 28 November 2013
Academic Editors: V. Bovdi, S. Dascalescu, V. Drensky, P. Koshlukov, and S. Yang
Copyright © 2013 M. R. Khan and M. M. Hasnain. Tis is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Tere has been considerable interest in the connection between the structure and the -structure of a ring, where denotes an
involution on a ring. In this context, Oukhtite and Salhi (2006) introduced a new class or we can say an extension of prime rings
in the form of -prime ring and proved several well-known theorems of prime rings for -prime rings. A continuous approach
in the direction of -prime rings is still on. In this paper, we establish some results for -prime rings satisfying certain identities
involving generalized derivations on -ideals. Finally, we give an example showing that the restrictions imposed on the hypothesis
of the various theorems were not superfuous.
1. Introduction
Troughout the paper, will denote an associative ring with
center (). For any ,∈, the symbol [,] stands for
the Lie product − and the symbol ∘ denotes the
Jordan product + . A ring is called 2-torsion free, if
whenever 2 = 0, with ∈, then =0. Recall that a ring
is prime if, for any , ∈ , = 0 implies =0 or
=0. A ring equipped with an involution is to be -
prime if = () = 0 ⇒ = 0 or =0. An example,
according to Oukhtite and Salhi [1], shows that every prime
ring can be injected in -prime ring and from this point of
view -prime rings constitute a more general class of prime
rings. An ideal is a -ideal if is invariant under ; that is,
() = . Note that an ideal may not be a -ideal. Let be a
ring of integers and =×. Consider a map :→
defned by ((,)) = (,) for all , ∈ . For an ideal
= ×{0} of , is not a -ideal of since () = {0}× ̸ =.
Several authors have studied the relationship between the
commutativity of a ring and the behavior of a special mapping
on that ring. In particular, there has been considerable inter-
est in centralizing automorphisms and derivations defned on
rings (see, e.g., [2–4], where further references can be found).
As defned in [5, 6], an additive mapping :→ is called
generalized derivation with associated derivation if
()= ()+ ()= ()+ (), ∀,∈.
(1)
Familiar examples of generalized derivations are derivations
and generalized inner derivations and later included lef
multiplier, that is, an additive mapping :→ satisfying
() = () for all , ∈ . Since the sum of two
generalized derivations is a generalized derivation, every map
of the form ()=+(), where is a fxed element of
and , a derivation of , is a generalized derivation, and if
has 1, all generalized derivations have this form.
In 2006, Oukhtite and Salhi [1] introduced a new class
or we can say an extension of prime rings in the form of
-prime ring. However, the actual motivation behind their
frst successful work came from Posner’s [7] second theorem
only. In [8], they successfully extended the result for -prime
ring. Recently, a major breakthrough has been achieved by
Oukhtite and Salhi [9], where the important results by Posner,
Herstein, and Bell have been proved for -prime rings. More
precisely, Posner’s second theorem of existence of a nonzero
centralizing derivation on prime ring which makes the ring