Algebras defined by homomorphisms

Let \mathcal{R} be a  commutative ring with identity, let A and B be two \mathcal{R}-algebras and \varphi:B\longrightarrow A be an \mathcal{R}-additive algebra homomorphism. We introduce a new algebra A\times_\varphi B, and give some basic properties of this algebra. Generalized ۲-cocycle derivations on A\times_\varphi B are studied. Accordingly, A\times_\varphi B is considered from the perspective of Banach algebras.

Keywords:

algebra , cocycle , generalized derivation , Banach algebra