A Work on Generalized Reverse Derivation and Skew-Derivation on Prime Near-Rings

Abdu Madugu and Tasiu Abdullahi Yusuf
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Abdu Madugu: Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Umaru Musa Yar’adua University, Katsina, Nigeria
Tasiu Abdullahi Yusuf: Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Umaru Musa Yar’adua University, Katsina, Nigeria

International Journal of Research and Innovation in Social Science, 2025, vol. 9, issue 4, 5227-5232

Abstract: This paper investigates the analysis of prime near-ring by explore the detailed structure of the generalized derivations that satisfy some specific assumptions. Let N be a prime near-rings and G be a generalized reverse derivative associated with mapping d on N. An additive mapping d:N→N is said to be a derivation on N if d(xy)=d(x)y+xd(y) for all x,y ∈N. A mapping G: N→N associated with derivation d is called a generalized derivation on N if G(xy)=G(x)y+xd(y) for all x,y∈N. Also, a mapping d: N→N is said to be a reverse derivation on N if d(xy)=d(y)x+yd(x) for all x,y ∈N and a mapping G:N→N associated with reverse derivation d is said to be a generalized reverse derivation on N if G(xy)=G(y)x+yd(x) for all x,y∈N. We prove some results on commutativity of prime near-rings involving generalized reverse derivations. In addition, we prove that; for prime near-rings N, if d(x)d(y)±xy=0 for all x,y∈N then d=0 where d is a skew- derivation associated with an automorphism β∶ N→N. Furthermore, for a prime near-ring N with generalized derivative G associated with mapping d on N, if G(x)G(y)±xy=0 for all x,y∈G then d=0.

Date: 2025
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