Is $\mathbb{Z}_2$ an injective module over $\mathbb{Z}_4$ and $\mathbb{Z_6}$? Is $\mathbb{Z}_2$ an injective module over $\mathbb{Z}_4$ and $\mathbb{Z_6}$?
I don't know which definition should I use to answer this question. Maybe I should use that $\mathbb{Z}_6 \equiv \mathbb{Z}_2 \oplus \mathbb{Z}_3$ or should I approach it with Baer's criterion?
Thanks in advance for any help.

$\mathbb Z_6$ is a semisimple ring, so all of its modules are injective. This is easy to see since it is isomorphic to the product of two fields. Using the Baer criterion immediately shows you why all modules are injective.
$\mathbb Z_2$ isn't an injective module of $\mathbb Z_4$ for the following reason: $\mathbb Z_4$ does not have any nontrivial summands, but it contains a copy of $\mathbb Z_2$ in the form of $(2+4\mathbb Z)$. But an injective module must split out of every module that contains it, and it clearly doesn't split out of $\mathbb Z_4$, so it's not injective.
