DOI 10.1515/gmj-2015-0016 | Georgian Math. J. 2015; aop Research Article Shakir Ali, Nadeem Ahmed Dar and Mustafa Asci On derivations and commutativity of prime rings with involution Abstract: In [6], Bell and Daif proved that if is a prime ring admitting a nonzero derivation such that () = () for all , ∈ , then is commutative. The objective of this paper is to examine similar prob- lems when the ring is equipped with involution. It is shown that if a prime ring with involution ∗ of a characteristic different from 2 admits a nonzero derivation such that ( ∗ ) = ( ∗ ) for all ∈ and () ∩ () ̸ = (0), then is commutative. Moreover, some related results have also been discussed. Keywords: Prime ring, normal ring, involution, derivation MSC 2010: 16W10, 16N60, 16W25 || Shakir Ali, Nadeem Ahmed Dar: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India, e-mail: shakir.ali.mm@amu.ac.in, ndmdarlajurah@gmail.com Mustafa Asci: Department of Mathematics, Pamukkale University, Denizli 20100, Turkey, e-mail: mustafa.asci@yahoo.com 1 Introduction This research has been motivated by our earlier work [1]. Throughout this article, will represent an associa- tive ring with centre (). For , ∈ , [, ] will be the element − and ∘ the element + , respec- tively. However, given two subsets and of , then [, ] will denote the additive subgroup of generated by all elements of the form [, ] where ∈ and ∈ and ∘ is defined similarly. Further, will be the subring of generated by . A ring is said to be 2-torsion free if 2 = 0 (where ∈) implies =0. A ring is called a prime ring if = (0) (where , ∈ ) implies =0 or = 0, and is called a semiprime ring in case = (0) implies =0. An additive map → ∗ of into itself is called an involution if (i) () ∗ = ∗ ∗ and (ii) ( ∗ ) ∗ = hold for all , ∈ . A ring equipped with an involution is called a ring with involution or a ∗-ring. An element in a ring with involution ∗ is said to be hermitian if ∗ = and skew-hermitian if ∗ = −. The sets of all hermitian and skew-hermitian elements of will be denoted by () and (), respectively. If is 2-torsion free then every ∈ can be uniquely represented in the form 2 = ℎ + where ℎ ∈ () and ∈ (). Note that in this case is normal, i.e., ∗ = ∗ , if and only if ℎ and commute. If all elements in are normal, then is called a normal ring. An example is the ring of quaternions. A description of such rings and further references can be found in [12]. An additive mapping :→ is said to be a derivation of if () = () + () for all , ∈ .A derivation is said to be inner if there exists ∈ such that () = − for all ∈. Over the last 30 years, several authors have investigated the relationship between the commutativity of the ring and certain special types of maps on . The first result in this direction is due to Divinsky [10] who proved that a simple artinian ring is commutative if it has a commuting non-trivial automorphism. Two years later, Posner [16] proved that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Over the last few decades, several authors have refined and extended these results in various directions (viz., [6, 7], where further references can be found). In the year 1995, Bell and Daif [6] showed that if is a prime ring admitting a nonzero derivation such that () = () for all , ∈ , then is commutative. This result was extended for semiprime rings in [8] by Daif. Further, for semiprime rings, Andima and Pajoohesh [3] showed that an inner derivation satisfying the above mentioned condition on a nonzero ideal of must be zero on that ideal. Moreover for semiprime rings with identity, they generalized this result to inner derivations of powers of and in [3]. Recently, many Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 7/12/15 2:22 PM