A study on orthogonal generalized derivations in semirings

Abstract

The Concept of semiring was first introduced by Vandiver in 1934. newlineHowever, the developments of the theory in semirings have been taking place newlinesince 1950. Semirings abound in the Mathematical World around us. A newlinesemiring is one of the fundamental structure in Mathematics. Indeed, the first newlinemathematical structure we encounter the set of natural numbers is a semir- newlineing. newlineIn the present work, we would like to study the structure of an orthog- newlineonality in a semiprime semirings having a various derivations which satis- newlinefies suitable algebraic properties. Orthogonal Generalized Derivations, Or- newlinethogonal Semiderivations, Orthogonal Generalized Reverse Derivations, Or- newlinethogonal Reverse Semiderivations, Orthogonal Symmetric Biderivations and newlineLeft Biderivations on Semiprime Semirings are characterized and also inves- newlinetigated that Generalized Derivations, Semiderivations, Reverse Derivations, newlineReverse Semiderivations are Orthogonal. newlineHowever, based on all the above concept and applications of semirings, newlinewe introduced the notion of Orthogonal Reverse Derivations, Orthogonal Re- newlineverse Semiderivations, Orthogonal Symmetric Biderivations and Left Bideriva- newlinetions in Semiprime Semirings. Also, we investigated Orthogonality between newlinethe Derivations and Biderivations in semiprime semirings. The Necessary newlineand Sufficient conditions between two Reverse Derivations be Orthogonal, newlinetwo Reverse Semiderivations be Orthogonal, two Symmetric Biderivations newlineto be Orthogonal. Also, we investigated if S is a semiprime semiring and B newlineis a Left Biderivation, then B must be an ordinary Biderivation that maps S newlineinto its centre newline