On (σ, τ)-module extension Banach algebras

Let A be a Banach algebra and X be a Banach A-bimodule. In this paper, we define a new product on A ⊕ X and generalize the module extension Banach algebras. We obtain characterizations of Arens regularity, commutativity, semisimplity, and study the ideal structure and derivations of this new Banach algebra.