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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?
Consider the subgroup of $(R,+)$ generated by the identity $1$. By hypothesis this is an ideal. But an ideal containing $1$ is the entire Ring. That means that each $r\in R$ has the form $n\cdot 1$.
Now there are two cases: Either there is a minimal $n\in \Bbb N$ such that $n\cdot 1=0$ (this is called the characteristic of $R$), or $R$ has infinitely many elements of the form $k\cdot 1,\ k\in \Bbb Z$
Hint: $\{n\cdot 1_R: n\in\mathbb Z\}$ is an ideal of $R$.
Hint
The group $\langle 1\rangle$ is an ideal so for $a\in A,\ a.1\in \langle 1\rangle $.
