Product of divisible module is divisible

Question

I have the following problem

Is the product of divisible $R$-modules divisible?

I think it is not. But I need some counterexample to this, somebody can give me a "place" to search one?

Thanks a lot!

Answer 1

Suposse that $\{M_\lambda\}_{\lambda\in\Lambda} $ is family of divisible $R$-modules and consider $Z=\prod_{\lambda\in\Lambda} M_\lambda$. Take $r\in R$ and $(x_\lambda)_{\lambda\in\Lambda}\in Z$, we can chose for each $\lambda$ an element $y_\lambda$ such that $r y_{\lambda}=x_\lambda$ then we have that $r (y_\lambda)_{\lambda\in\Lambda}=(x_\lambda)_{\lambda\in\Lambda}$. With this we prove that $Z$ is divisible.