arXiv:2002.03101v1 [math.RA] 8 Feb 2020 On ∗−Reverse Derivable Maps Gurninder S. Sandhu Department of Mathematics, Patel Memorial National College, Rajpura 140401, Punjab, India. e-mail: gurninder rs@pbi.ac.in Bruno L. M. Ferreira Technological Federal University of Paran´ a, Professora Laura Pacheco Bastos Avenue, 800, 85053-510, Guarapuava, Brazil. e-mail: brunoferreira@utfpr.edu.br and Deepak Kumar Department of Mathematics, Punjabi University, Patiala-147002, Punjab, India. e-mail: deep math1@yahoo.com Abstract Let R be a ring with involution containing a nontrivial symmetric idempotent element e. Let δ : R → R be a mapping such that δ(ab)= δ(b)a * + b * δ(a) for all a, b ∈ R, we call δ a *-reverse derivable map on R. In this paper, our aim is to show that under some suitable restrictions imposed on R, every *-reverse derivable map of R is additive. 2010 Mathematics Subject Classification. 17C27 Keyword: Additivity, Reverse derivable maps, Involution, Peirce decomposi- tion. 1 Introduction Let R be a ring, by a derivation of R, we mean an additive map δ : R → R such that δ(ab)= δ(a)b + aδ(b) for all a,b ∈ R. A derivation which is not necessarily additive is said to be a multiplicative derivation or derivable map of R. A mapping δ : R → R is known as multiplicative Jordan derivation of R if δ(ab + ba)= δ(a)b + aδ(b)+ δ(b)a + bδ(a) for all a,b ∈ R. In addition, δ is called n−multiplicative derivation of R if δ(a 1 a 2 ··· a n )= ∑ n i=1 a 1 a 2 ··· δ(a i ) ··· a n for all a 1 ,a 2 , ··· ,a n ∈ R. In [12], Herstein introduced a mapping “ † ” satisfying (a + b) † = a † + b † and (ab) † = b † a + ba † called a reverse derivation, which is certainly not a derivation. Moreover, a mapping δ : R → R satisfying δ(ab)= δ(b)a+bδ(a) for all a,b ∈ R is called a multiplicative reverse derivation or reverse derivable map of R. Let e be an idempotent element of R such that e =0, 1. Then R can be decomposed as follows: R = eRe  eR(1 − e)  (1 − e)Re  (1 − e)R(1 − e) 1