I would like to get an example of a finite commutative ring $R$ and a subring $S$ of $R$ that is not an ideal.
I have tried working with $\mathbb Z_n$ and most of the examples I have tried end up being either both or none. Can we say that every subring of $\mathbb Z_n$ is also an ideal? (I think the fact that all subrings of $\mathbb Z$ look like $k\mathbb Z$ for some $k$ which are also the only ideals of $\mathbb Z$ probably factors in here)
Is a result like true in general? That is, is any ideal of a finite commutative ring also a subring?