  Let $A$ and $B$ be Banach algebras and let $B$ be an algebraic Banach
$A-$bimodule. Then the $\ell^1-$direct sum $A\times B$ equipped with the
multiplication $$(a_1,b_1)(a_2,b_2)=(a_1a_2,a_1\cdot b_2+b_1\cdot
a_2+b_1b_2),~~ (a_1, a_2\in A, b_1, b_2\in B)$$ is a Banach algebra denoted by
$A\bowtie B$. Module extension algebras, Lau product and also the direct sum of
Banach algebras are the main examples satisfying this framework. We obtain
characterizations of bounded approximate identities, spectrum, and topological
center of this product. This provides a unified approach for obtaining some
known results of both module extensions and Lau product of Banach algebras.
