Orthogonal Jordan Derivations on $\Gamma$-semihyperrings

Resumo

The present paper introduces the concept of orthogonal derivation on $\Gamma$-semihyperrings and explores some fundamental properties of orthogonal Jordan derivation. It is shown that a non-zero derivation is not necessarily orthogonal to itself and the specific condition under which such a derivation becomes orthogonal to itself is established. Furthermore, the sum of two orthogonal derivations remains orthogonal to each of its summands is proved. A necessary and sufficient condition for two derivations to be orthogonal is also derived. The study concludes with an in-depth examination of orthogonal Jordan derivations, revealing important results regarding their behaviour in the context of $\Gamma$-semihyperrings.