Factorization property and Arens regularity

In this paper, we study the Arens regularity properties of module actions and we extend some proposition from Baker, Dales, Lau and others into general situations. For Banach $A-bimodule$ $B$, let $Z_1(A^{\ast \ast })$, $Z_{B^{\ast \ast }}^{\mathrm{\ell }}(A^{\ast \ast })$ and $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$ be the topological centers of second dual of Banach algebra $A$, left module action ${\pi }_{\mathrm{\ell }}:A\times B\to B$ and right module action $\pi_r:~B\times A\rightarrow B$, respectively. We establish some relationships between them and factorization properties of $A^{\ast }$ and $B^{\ast }$. We search some necessary and sufficient conditions for factorization of $A^{\ast }$, $B$ and $B^{\ast }$ with some results in group algebras. We extend the definitions of the left and right multiplier for module actions.