# The ring S is integral domain or field?

Let $$R=\mathbb{Z}[i] := \{a + ib \in \mathbb{ C} : a, b \in \mathbb{Z}\}$$ be the ring of Gaussian integers. Let $$I := (4 + 5i)$$ be the principal ideal generated by $$4 + 5i$$ and $$S := \mathbb{Z}[i]/ I$$ be the quotient ring. Then

• $$S$$ is an integral domain.
• $$S$$ is not an integral domain.
• $$S$$ is a field.
• $$S$$ is an integral domain but not a field

I think this will be integral domain because $$\Bbb{Z}$$ is integral domain but not field.

Any hints/ solution

Thanks u

• $\Bbb Z[i]/(4+5i)\cong \Bbb Z / 41 \Bbb Z$, so $S$ is a field – Mustafa Oct 9 '18 at 4:10

$$4+5i$$ is irreducible in $$\Bbb{Z}[i]$$, since $$\vert \vert 4+5i \vert \vert=4^2+5^2=41$$, a prime number
So $$\langle 4+5i \rangle$$ is maximal and $$R/I$$ is a field as well as an integral domain