Find a subring of $\Bbb Z \oplus \Bbb Z$ that is not an ideal of $\Bbb Z \oplus \Bbb Z$. Find a subring of $\Bbb Z \oplus \Bbb Z$ that is not an ideal of $\Bbb Z \oplus \Bbb Z$.
I can't see any way a subring of $\Bbb Z \oplus \Bbb Z$ can NOT be an ideal.  Subrings of $\Bbb Z \oplus \Bbb Z$ are of the form $n\Bbb Z \oplus k\Bbb Z$ where $n$ and $k$ are integers.  So for something to not be an ideal in $\Bbb Z \oplus \Bbb Z$ there must be an $x = (j_1, j_2)$ in $\Bbb Z \oplus \Bbb Z$ and an $y = (b_1n, b_2k)$ in $n\Bbb Z \oplus k\Bbb Z$ such that $xy$ or $yx \notin$ $n\Bbb Z \oplus k\Bbb Z$.  But this doesn't make sense because it seems any integer pair can be reached.
Anyone have any ideas?

Any subring would I presume contain $(1,1)$, the ring's unity (this is not the only possible interpretation).
Then any proper subring will do.
The proper subrings are the $R_n=\{(a,b)\in\Bbb Z^2\mid a\equiv b\pmod n\},$ for $n\gt1.$  See @Qiaochu's proof here.
$R_\infty$ is @Arthur's example.
