  Let $T$ be a Banach algebra homomorphism from a Banach algebra $\mathcal B$
to a Banach algebra $\mathcal A$ with $\|T\|\leq 1$. Recently it has been
obtained some results about Arens regularity and also various notions of
amenability of $\mathcal A\times_{T}\mathcal B$, in the case where $\mathcal A$
is commutative. In the present paper, most of these results have been
generalized and proved for an arbitrary Banach algebra $\mathcal A$.
