Classification of additive mappings on certain rings and algebras

The objective of this research is to prove that an additive mapping \(\Delta :{\mathcal {A}}\rightarrow {\mathcal {A}}\) will be a generalized derivation associated with a derivation \(\partial :{\mathcal {A}}\rightarrow {\mathcal {A}}\) if it satisfies the following identity \(\Delta (r^{m+n+p})=\Delta (r^m)r^{n+p}+r^m\partial (r^{n})r^p+r^{m+n}\partial (r^p)\) for all \(r\in {\mathcal {A}}\), where \(m, n\ge 1\) and \(p\ge 0\) are fixed integers and \({\mathcal {A}}\) is a semiprime ring. Another analogous has been done where an additive mapping behaves like a generalized left derivation associated with a left derivation on \({\mathcal {A}}\) satisfying certain algebraic identity. The proofs of these advancements are derived employing algebraic concepts. These theorems have been validated by offering an example that shows they are not insignificant. Furthermore, we provide an application in the framework of Banach algebra.