The k-Derivation Acting as a k-Endomorphism and as an Anti-k-Endomorphism on Semiprime Nobusawa Gamma Ring

Abstract

ABSTRACT

$\Gamma^*$-DERIVATION ACTING AS AN ENDOMORPHISM AND AS AN ANTI-ENDOMORPHISM IN SEMIPRIME $\Gamma$-RING M WITH INVOLUTION

“…In Γ-rings, Dey and Paul [4] proved that if D is a generalized derivation of a prime Γ-ring M with an associated derivation d of M which acts as a homomorphism and an anti-homomorphism on a non-zero ideal I of M, then d = 0 or M is commutative. Afterwards, Chakraborty and Paul [11] worked on kderivation of a semiprime Γ-ring in the sense of Nobusawa [10] and proved that d = 0 where d is a k-derivation acting as a k-endomorphism and as an anti-kendomorphism, the above mentioned results following [6] in classical rings are extended to those in gamma rings with derivation acting as a homomorphism and as an anti-homomorphism on σ-prime Γ-rings. In this paper we will prove that if d is Γ * -derivation of a semiprime Γ-ring with involution which is either an endomorphism or anti-endomorphism, then d=0.…”