Showing $M_1$ and $M_2$ are submodules of $M$, where $M$ is a module of a direct product algebra.

Question

Let $A = A_1 \times A_2$, the product of two $k$-algebras.

Suppose $M$ is some A-module.

Define $M_1 \triangleq \{(A_1,0)m : m \in M\},\ M_2 \triangleq \{(0,A_2)m : m \in M \}$.

Show that $M_1$ and $M_2$ are submodules of $M$ and that $M = M_1 \oplus M_2$

I'm very stuck on this, please help.