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

Abstract

This article defines k-endomorphism and anti-k-endomorphism on ?N-rings, and uses the concept of k-derivation of ?N-rings. Considering M as a semiprime ?N-ring and d as a k-derivation of M, it aims to prove that (i) if d acts as a k-endomorphism on M such that M?M=M and xk(?)x=0 for all x?M and ?? ?, then d=0; and (ii) if d is acting as an anti-k-endomorphism on M such that M?M=M, xk(?)x=0 and k(?)x?=?xk(?) for all x?M and ?? ?, then d=0.