I have worked out that $Z[i]/\langle 3+i \rangle = \{[0],[1],[2],...,[9]\}$. Now I want to state the ring operations but I am confused between it being “addition/multiplication modulo 10 or modulo $\langle 3+i \rangle$.
Please would someone share some light on this.
I am thinking to write $[x]+[y]=[x+y \mod10]$
And similar for multiplication.
Is this correct?
Consider the map $$ \varphi: \mathbb{Z} [x] \to \mathbb{Z} $$
defined by $\varphi(1) = 1, \varphi(x) = -3$, let $I$ be the ideal generated by $x^2 + 1$, then $\varphi(I) \subseteq 10 \mathbb{Z}$, so this map descends to a map $$ \varphi: \mathbb{Z}[i] = \mathbb{Z} [x]/I \to \mathbb{Z}/10\mathbb{Z}. $$ Here $i$ denotes the class of $x$. This map is clearly surjective as $\varphi(1) = 1$, and it is injective. Indeed if $x = a + bi$ is in the kernel then $a - 3b = 0 \mod 10$, then $x = b(3 + i) + 10k$ for some integer $k$, so $x \in I + 10\mathbb{Z}$ but $10\mathbb{Z} \subseteq I$ since $10 = (3 + i)(3-i)$. This gives your isomorphism, then the adition and multiplications are just the usual ones in $\mathbb{Z}/10\mathbb{Z}$, e.g. $[3] + [8] = [11] = [1]$.
Does this answers your question?