Example in Modules

Question

I am looking for an example of a ring where $M_1$ and $M_2$ are two Modules over that ring but their union isn't.

I was thinking of using the fact that any group is a module over $\mathbb{Z}$, and that $\mathbb{Q}$ is also a module over $\mathbb{Z}$ but I'm not sure.

Thanks.

Answer 1

What about taking any field $F$ and$$M_1=\{(x,0)\,|\,x\in F\}\text{ and }M_2=\{(0,x)\,|\,x\in F\}?$$