Subring and ideal structure of rings in the form $Z_n\oplus Z_m$

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So I was wondering how the ideals and subrings of something like this look like. I know how the structure of $Z_n$ looks like since every subring of a cyclic group is an ideal. However, this is clearly not cyclic unless $n$ is relatively prime with $m$. Any help would be helpful.

Thank you.