A Study on Special Kinds of Derivations in Ordered Hyperrings.

In this study, we concentrate on an important class of ordered hyperstructures with symmetrical hyperoperations, which are called ordered Krasner hyperrings, and discuss strong derivations and homo-derivations. Additionally, we apply our results on nonzero proper hyperideals to the study of derivations of prime ordered hyperrings. This work is a pioneer in studies on structures such as hyperideals and homomorphisms of an ordered hyperring with the help of derivation notation. Finally, we prove some results on 2-torsion-free prime ordered hyperrings by using derivations. We show that if d is a derivation of 2-torsion-free prime hyperring R and the commutator set [ l , d (q) ] is equal to zero for all q in R, then l ∈ Z (R) . Moreover, we prove that if the commutator set (d (l) , q) is equal to zero for all l in R, then (d (R) , q) = 0 . [ABSTRACT FROM AUTHOR]