[Q-1] Can we find subrings in $\mathbb{Z}$ that are not ideals of $\mathbb{Z}$?
Edit: There was another question, that I was trying to answer:
[Q-2] Find a subring of $\mathbb{Z}\oplus\mathbb{Z}$ that is not an ideal.
While solving [Q-2], It came to my mind, whether we can find any subring in $\mathbb{Z}$ that is not an ideal of $\mathbb{Z}$. Now, the first thing to do was find out subrings of $\mathbb{Z}$ and some of the examples are the set $n\mathbb{Z}$ under the normal operation. But can we find other subrings of $\mathbb{Z}$ itself?
The additive subgroups of $\mathbb{Z}$ are precisely $n\mathbb{Z}$ for $n\in{\mathbb{N_0}}$. Every such subgroup is also an ideal, therefore the answer to your question is (if I understood it correctly):
No. Every subring of $\mathbb{Z}$ is also an ideal.