Determine the ideal $\langle 2i \rangle$ of the Gaussian integers $\Bbb Z[i]$. Describe the quotient $\Bbb Z[i]/\langle 2i\rangle$.

Question

Determine the ideal $\langle 2i \rangle$ of the Gaussian integers $\Bbb Z[i]$. Describe the quotient $\Bbb Z[i]/\langle 2i\rangle$.

I am having confusion with the theorem for $\langle 2i \rangle$. It's proved in my course book that for a ring $R$ and $a \in R$ the ideal $\langle a \rangle = \{ra \mid r \in R\}$. But I've also seen that $\langle a \rangle = \{na \mid n \in \Bbb Z\}$.

So either $$\langle 2i \rangle = \{2ri \mid r \in \Bbb Z[i]\}=\{2(a+bi)i \mid a+bi \in \Bbb Z[i]\}=\{2ai -2b \mid a+bi \in \Bbb Z[i]\} = \{2(ai-b) \mid a+bi \in \Bbb Z [i]\}$$ or $$\langle 2i \rangle = \{2ni \mid n \in \Bbb Z\}$$

which one is it here?

Answer 1

You should check whatever course or book you are using for the definition, but the standard one (for a commutative ring $R$) is the former: $\left\langle a\right\rangle=\left\{ra\mid r\in R\right\}$. If $R=\mathbb{Z}$, this is of course equivalent to the latter.