I have a question that I can't seem to figure out:
Let $R$ be a commutative ring with identity. Note that under addition $R$ is an abelian group. Suppose every subgroup of this group is an ideal. Show that $R$ is isomorphic to the ring of integers or to the integers mod $n$.
I realize $\Bbb Z$ has such a property, and that if $R$ is infinite, we are in case 1, and if it is finite, then case 2, but despite a bunch of manipulating I can't get a sufficient proof working. Any help?