Orthogonal Generalized Symmetric Higher bi-Derivations on Semiprime Г-Rings .
Authors: Salah Mahdi Salih, Samah Jaber Shaker
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In this paper a Г-ring M is presented. We will study the concept of orthogonal generalized symmetric higher bi-derivations on Г-ring. We prove that if M is a 2-torsion free semiprime Г-ring , D_n and G_n are orthogonal generalized symmetric higher bi-derivations associated with symmetric higher bi-derivations d_n and g_n respectively for all n ϵN. Then the following relations are hold for all x,y,zϵM,αϵГ and nϵN:
(i) D_n (x,y)αG_n (y,z)=G_n (x,y)αD_n (y,z) =(0) hence D_n (x,y)α〖 G〗_n (y,z)+
G_n (x,y)αD_n (y,z)=0 .
(ii) d_n and G_n are orthogonal and d_n (x,y)αG_n (y,z)=G_n (x,y)αd_n (y,z)=(0) .
(iii) g_n and D_nare orthogonal and g_n (x,y)α D_n (y,z)=D_n (x,y)αg_n (y,z)=(0) .
〖(iv) d〗_n and g_n are orthogonal symmetric higher bi-derivations .
(v) d_n G_n=G_n d_n=0 and g_n D_n=D_n g_n=0 .
(vi) G_n D_n=D_n G_n=0 .