  Let $\cal A$ be a Banach algebra. Then $\cal A^{**}$ the second dual of $\cal
A$ is a Banach algebra with first (second) Arens product. We study the Arens
products of $\cal A^{4}(=({\cal A^{**}})^{**}).$ We found some conditions on
$\cal A^{**}$ to be a left ideal in $\cal A^{4}.$ We found the biggest two
sided ideal $I$ of ${\cal A},$ in which $I$ is a left (right) ideal of ${\cal
A}^{**}$.
