Does the non-zero ideal $I=\langle a+ib\rangle$ contain no positive integers?

Question

I have a question as follows given by my professor:

Does the non-zero ideal $I=\langle a+bi\rangle$ contain no positive integers?

I answered as follows:

Since, $I$ is non-zero so, $a+bi$ is non-zero and since $a-bi$ belong to $\Bbb{Z}[i]$ so $(a+bi)(a-bi) \in I$ implies that $a^2+b^2 \in I$, which is a positive integer. So $I$ contains positive integers.

Is my explanation correct?

Answer 1

Your explanation is entirely correct. I assume that it is clear from the context that $I=\langle a+bi\rangle$ is supposed to be an ideal of $\Bbb{Z}[i]$, and that $a$ and $b$ are supposed to be integers.