Let $A_1,A_2$ be two rings and $e_1=(1_{A_1},0_{A_2})$ and $e_2=(0_{A_1},1_{A_2})$. Note that $e_1\cdot (A_1\times A_2)$ and $e_2\cdot (A_1\times A_2)$ are two-sided ideals of $A_1\times A_2$ as they are equal to the kernels of the projections $pr_2,pr_1$, respectively.
Let $E$ be an $(A_1\times A_2)$-module. Then $e_1E$ and $e_2E$ are submodules of $E$. On the other hand, $e_1$ annihilates $e_2E$ and $e_2$ annihilates $e_1 E$.
How can I use this data to define an $A_1$-module structure on $e_1E$ and an $A_2$-module structure on $e_2E?$
Attempt:
I know that $A_1\simeq(A_1\times A_2)/e_2(A_1\times A_2)$ and $A_2\simeq(A_1\times A_2)/ e_1(A_1\times A_2)$. This means that I need to find $e_1(A_1\times A_2)$-module structure on $e_2E$ and $e_2(A_1\times A_2)$-module structure on $e_1 E$. I don't know what these structures are. They are probably derived by using the annihilations but I don't know exactly how.
Apparently you are working with left modules.
There is basically nothing to do.
$r\cdot (e_1m):=(re_1)\cdot m$ for $r\in A_1$ and $m\in E$ gives the obvious $A_1$ module structure to $e_1E$
and
$r\cdot (e_2m):=(re_2)\cdot m$ for $r\in A_2$ and $m\in E$ gives the obvious $A_2$ module structure to $e_2E$
Perhaps the only thing is to observe they are unital module structures since $e_1$ is the identity of $A_1$ and $e_2$ is the identity of $A_2$.