Orthogonal Generalized Higher Symmetric Reverse Bi-Derivations on Semiprime Γ- Rings

Abstract

The purpose of this paper is to study the concept of orthogonal generalized

higher symmetric reverse bi- derivation on semiprime Γ-ring. We study some

lemmas and theorems of orthogonality on semiprime Γ-rings. We prove that

if M is a 2-tortion free semiprime Γ-ring then Dn and Gn are orthogonal

generalized higher symmetric reverse bi-derivations associated with higher

symmetric reverse bi-derivations dnand gn for all n∈N. Then the following

relations are hold for all a,b,c∈M , α ∈ Γ and n∈N:

i. Dn

(a, b)αGn

(b, c) = Gn

(b, c)αDn(a, b) = 0

hence Dn(a, b)αGn(b, c) + Gn(b, c)αDn(a, b) =0

ii. dn and Gn orthogonal and dn(a, b)αGn(b, c)=Gn(b, c)αdn(a, b) = 0.

iii. gn and Dn orthogonal and gn(a, b)αDn(b, c)=Dn(b, c)αgn(a, b) = 0.

iv. dnGn = Gndn = 0 and gnDn = Dngn = 0

v. GnDn = DnGn = 0.