A class of solutions for functional equations originated from the product in a quartic number field

Let ???? be a semigroup and let ???? be a unital algebra over a field R. The purpose of this research is to find the solutions ???? : ????^4 → ???? for the functional equation ???? (????_1, ????_1, ????_1, ????_1) ???? (????_2, ????_2, ????_2, ????_2)= ???? (????_1????_2 + ????????_1????_2 + ????????_1????_2 + ????????_1????_2, ????_1????_2 + ????_1????_2 + ????????_1????_2 + ????????_1????_2, ????_1????_2 + ????_1????_2 + ????_1????_2 + ????????_1????_2, ????_1????_2 + ????_1????_2 + ????_1????_2 + ????_1????_2)for all (????_1, ????_1, ????_1, ????_1), (????_2, ????_2, ????_2, ????_2) ∈ ????^4, where ???? ∈ R, which arises from the formula for the product of two numbers in the quartic number field Q(4√????). Our results are a generalization of the recent result in [Mouzoun, A., Zeglami, D. (2022). A classof semigroup homomorphisms arising from the product in a pure quartic number field. Boletín de la Sociedad Matemática Mexicana, 28(3), 73.]. Moreover, we will improve the solutions ???? : R^4 → ???? of the aforementioned research article by presenting the new procedure to fix the mistake for such results.