Suppose $A = A_1 \times A_2$ where $A_1$ and $A_2$ are associative algebras (Not necessarily unitary). Show that any $A$-Module $T$ is isomorphic to $M \oplus N$ where $M$ is an $A_1$-module and $N$ is an $A_2$-module.
What we have so far is using the fact that $(a_1,a_2) = (a_1,0) + (0,a_2)$. We can get $(a_1,0)T = T_1$ and $(0,a_2)T = T_2$ are both submodules of $T$. So $T = T_1 + T_2$.
We are having problems showing that $T_1 \cap T_2$ is trivial. Once we have that fact then $T_1$ and $T_2$ are isomorphic to $M$ and $N$ respectively.
Any help would be very much appreciated.
Note you can define $T_1$ as the set of elements in $T$ fixed by $(1, 0) \in A_1 \times A_2$ and you can define $T_2$ as the set of elements fixed by $(0, 1)$. If you're fixed by both $(1, 0)$ and $(0, 1)$ then you're fixed by their product.