Jordan Derivation on the Polynomial Ring R[x]

Given a ring R. An additive mapping δ: R → R is called a Jordan derivation if δ(a²) = δ(a)a + aδ(a) for every a in R. Jordan derivation is one of the special forms of derivation. In this study, we investigate the Jordan derivation on the polynomial ring R[x] and examine its properties. This study begins by constructing the Jordan derivation on the polynomial ring R[x], followed by investigating its characteristics, including the relationship between the Jordan derivation on the ring R and on the polynomial ring R[x]. In addition, several concrete examples are presented to illustrate the main results obtained. This research is expected to contribute to a deeper understanding of the properties of Jordan derivations on polynomial rings.