Let $\{R_n\}_{n\in \mathbb{N}}$ be a collection of fields. Let $R:=\prod\limits_{n=1}^{\infty}R_n$. Show that for every $n$ $R_n$ is an injective $R$-module, but $\bigoplus\limits_{n=1}^{\infty}R_n$ is not injective.
Edit: First part is a consequence of Baer's criterion, but I don't see how to do the second part.
On
It is clear that the product is injective wrt itself and from a theorem that a product is injective iff each factor is injective the first part is settled. As for the second part we need to see that we cannot inject the product inside the direct sum since we have infinitely many factors, see Ribenbenboim : Rings and modules top of page 25.
The inclusion $\bigoplus R_n \hookrightarrow \prod R_n$ is essential, hence $\bigoplus R_n$ is not injective.