Show every subring of ring $\Bbb Z_n$ is ideal There was a similar question on this website but it was for $\Bbb Z$. $\Bbb Z$ is not the same as $\Bbb Z_n$ is it?
However, I said let $S$ be a subring of $\Bbb Z_n$. Then $S$ is a ring and $(S,+)$ is a group. Let $m\in \Bbb Z_n$and $s$ be in $S$. Then: $s+s+..+s (\text{m times}) = sm$. Then $sm \in S$.
Would this be sufficient? Or do we need to break this down into cases of some sort? Thanks!

Let $S$ be a subring of $\Bbb Z_n$. 
(Since  $\Bbb Z_n $ is cyclic so will be every subgroup of $\Bbb Z_n$).
Then $S=\langle [a]\rangle =\{n[a]:n\in \Bbb  Z\}$ 
Now take $[r]\in \Bbb Z_n$,then $[r]=r[1]$.
Take any $[s]\in S$ .Then $[s]=p[a]$.
Then $[rs]=r[1]p[a]=rp[a]$ where $rp\in \Bbb Z$ . Hence $[rs]\in S$.
Hence $S$ is an ideal of $\Bbb Z_n $.
