Example of reduced module

Question

An $R$-module $M$ is a reduced module if $M$ has no nonzero injective submodules.

I need example of a reduced module.

Answer 1

$\mathbf Z$, and more generally any P.I.D. which is not a field: its ideals are isomorphic to the ring, and an injective module over an integral domain is divisible – which would imply $\mathbf Z$ is a field.

Answer 2

If $R$ is any nonfield integral domain, then $R=M$ is a reduced module.

There are no proper injective submodules because they would have to split out of $R$, and be generated by an idempotent. But $R$ only has trivial idempotents, so the idempotent would have to be $1$. But an integral domain that is self-injective is a field.

Of course, you can say the same thing if you take $R$ a ring which only has trivial idempotents and is not right self-injective.

Here are some examples that aren't domains.