Determine the number submodules of $\mathbb{Z}_6 \times \mathbb{Z}_6$ as a $\mathbb{Z}_{12}$-module.
Today our professor introduced the notion of modules and gave this problem as an exercise. I have thought over it and I have come up with the following solution. Is this is correct?
My attempt:
The abelian group $\mathbb{Z}_6 \times \mathbb{Z}_6 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3$ has 16 subgroups. Now I know that "An abelian group $A$ is a $\mathbb{Z}_m$-module iff $mA=0$. From this I conclude that each of the subgroups of $\mathbb{Z}_6 \times \mathbb{Z}_6$ should also be a submodule. Hence there are 16 submodules.