Let $a,b$ be integers. I would to know what other ring is $R=\Bbb Z[X]/(aX+b)$ isomorphic to?
If $a$ is a unit of $\Bbb Z$, then $R \cong \Bbb Z$. If $a=0$, then $R \cong (\Bbb Z/b\Bbb Z)[X]$. If $a=2,b=0$, then $R \cong \Bbb Z \oplus \Bbb F_2[X]$ as abelian groups at least, but I'm not sure as rings. If $b$ is a multiple of $a$, we could use the Chinese remainder theorem, I think.
But in general, for instance $\Bbb Z[X]/(2X+3)$ or $\Bbb Z[X]/(6X+4)$ I don't know how to manage. It would also be interesting to know what happens if we replace $\Bbb Z$ by any other commutative ring (and $a,b$ elements of that ring)...