AN INNOVATIVE STUDY ON REVERSE DERIVATION IN NEARRINGS
European Journal of Molecular & Clinical Medicine,
2020, Volume 7, Issue 3, Pages 4865-4872
AbstractAn additive function f from a ring R to itself is known as a derivation if f(ab)=f(a)b+af(b) for all a,b in R.An additive function f from a ring R to itself is known as a reverse derivation if f(ab)=f(b)a+bf(a) for all a,b in R.An additive mapping d:R to R is called a right generalized derivation if there exists a derivation f:R to R satisfying d(ab)=d(a)b+af(b) for all a,b in R.An additive mapping d:R to R is called a left generalized derivation if there exists a derivation f:R to R such that d(ab)=f(a)b+ad(b) for all a,b in R.An additive map d:R to R is called a generalized derivation if it is both right and left generalized derivations.This research article mainly explores on reverse derivation in nearrings. The prime objective of this talk is to present an elegant proof for”Let M be a prime nearring with a nonzero reverse derivation d. If d (M) is a subset of Z, then (M, +) is an abelian.Moreover, if M is a two torsion free, then M is commutative ring, If d acts as a homomorphism on M and anti-homomorphism on M then d=0”. In order to present an innovative proof for this result five prerequisites in terms of lemmas are proposed. In this discourse we present some fundamental characteristic properties of reverse derivation in nearrings and some evolutions in the conjecture of reverse derivationin nearrings.The innovatory proofs proposed here ensures a way for young researchers to extend these ideas to Jordan generalized derivations in gamma nearrings.
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