Artin algebra problem This is Artin Algebra chapter 11 section 4 problem 3(b).
Identify  ring $\Bbb{Z}[i]/(2+i)$.
Solution: Artin didn't  explicitly mention what did he wants to ask? Anyways let me start.
Define a map $\psi:\mathbb{Z}[x]\to \Bbb Z[i]$ by $f\mapsto f(i)$.
Clearly $\psi$ is a surjective ring homomorphism with kernel generated by the ideal $(x^2+1)$.
Thus by 1st isomorphism theorem,
$$\frac{\Bbb Z[x]}{(x^2+1,x+2)}\cong \frac{\Bbb Z[i]}{(\psi(x^2+1),\psi(x+2))}=\frac{\Bbb Z[i]}{(2+i)}.$$
Define a map $\phi:\Bbb Z[x]\to\Bbb Z$ by $\phi(g)=g(-2)$.
Clearly $\phi$ defines a  surjective ring homomorphism with kernel generated by $(x+2)$.
Thus by 1st isomorphism theorem we have
$$\Bbb Z[x]/(x^2+1,x+2)\cong  \Bbb Z/(\phi(x^2+1),\phi(x+2))=\Bbb Z/5\Bbb Z. $$
Please verify this solution!
Thanks beforehand!
 A: Yes. Your solution is correct. What the exercise is asking you to do is to recognise the given ring as a "simpler" ring.
In fact, more generally, one has $$\frac{\Bbb Z[\iota]}{a + \iota} \cong \frac{\Bbb Z}{a^2 + 1}$$
for all $a \in \Bbb Z$. Your approach should work for this result as well.

Here's an alternate way that might feel more direct since the maps are only between the rings involved and not $\Bbb Z[x]$. (But the calculations are much cleaner in your method.)
Define $\varphi : \Bbb Z \to \Bbb Z[\iota]/(a + \iota)$ to be the composition of the natural inclusion and projection $$\Bbb Z \to \Bbb Z[\iota] \to \frac{\Bbb Z[\iota]}{a + \iota}.$$
First, show that $\varphi$ is onto. (Hint: show $\varphi(c - ad) = \overline{c + \iota d}$.)
Second, show that $\ker \varphi$ is precisely $(a^2 + 1)$. (This is also not too tough to show. To show the non-trivial inclusion, consider $n \in \ker \varphi$. Write $n = (a + \iota)(c + \iota d)$ and then deduce $c = -ad$ and conclude the result.)
