A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner’s result to generalized derivations centralizing on Jordan ideals of rings with involution and discuss the related results. Moreover, we provide examples to show that the assumed restriction cannot be relaxed. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 225 – 232 ¨ ITAK ˙ c TUB ⃝ doi: Generalized derivations centralizing on Jordan ideals of rings with involution Lahcen OUKHTITE∗, Abdellah MAMOUNI Algebra and Applications Group, Department of Mathematics, Faculty of Science and Technology, Universit´e Moulay Isma¨ıl Boutalamine, Errachidia, Morocco Received: • Accepted: • Published Online: • Printed: Abstract: A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner’s result to generalized derivations centralizing on Jordan ideals of rings with involution and discuss the related results. Moreover, we provide examples to show that the assumed restriction cannot be relaxed. Key words: Rings with involution, generalized derivations, Jordan ideals 1. Introduction Throughout R will represent an associative ring with center Z(R). For any x, y ∈ R, the symbol [x, y] stands for the commutator xy − yx and we will make use of the following basic commutator identities without any specific mention: [x, yz] = y[x, z] + [x, y]z, [xy, z] = x[y, z] + [x, z]y. R is 2 -torsion free if 2x = 0 yields x = 0. We recall that R is prime if aRb = 0 implies a = 0 or b = 0. A ring with involution (R, ∗) is ∗-prime if aRb = aRb∗ = 0 yields a = 0 or b = 0. Note that every prime ring having an involution ∗ is ∗ -prime but the converse is in general not true. For example, if Ro denotes the opposite ring of a prime ring R , then R × Ro equipped with the exchange involution ∗ex , defined by ∗ex (x, y) = (y, x), is ∗ex -prime but not prime. This example shows that every prime ring can be injected in a ∗-prime ring and from this point of view ∗ -prime rings constitute a more general class of prime rings. An additive subgroup J of R is said to be a Jordan .
