(untitled) (European Journal of Pure and Applied Mathematics, 2025)
The intention of the current investigation is to demonstrate that if an additive mapping H:R→ R fulfills any one of the following identities: enumerate [(i)] 3H(r^3p)=H(r^p)φ(r^2p)+φ(r^p) H(r^p)φ(r^p)+ φ(r^2p)H(r^p) [(ii)] 2H(r^2p)=H(r^p)φ(r^p)+φ(r^p)H(r^p) [(ii)] H(r^3p)=φ(r^p) H(r^p) φ(r^p) for all rR, enumerate then H is a φ-centralizer on R, where R is any suitable, torsion-free semiprime ring and p is a fixed integer greater than or equal to 1. As a result of the primary theorems, involution I_v related observations are also provided. We will also consider criticism and discussion alongside the proofs of theorems. Suitable examples given in favor of justification.