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Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is an ideal of $\Bbb Z[i]$. Is $I$ a maximal ideal? And to find the numbers of elements of the quotient ring $\Bbb Z[i]/ I$.

I have shown that $I$ is an ideal of $\Bbb Z[i]$. But how to verify that $I$ a maximal ideal or not?

And to find the numbers of elements of the quotient ring $\Bbb Z[i]/ I$.

Is there any general way to verify when $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $n\}$ will be maximal or not?

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    $\begingroup$ $I \subseteq R$ is a maximal ideal iff $R/I$ is a field. $I$ is prime iff $R/I$ is an integral domain. $\endgroup$ – Arthur Apr 17 '15 at 12:34
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$I$ is not prime since $(2+i)(2-i) = 5 \in I$, and $2+i,2-i \notin I$. In particular, $I$ is not a maximal ideal.

Finally, note that $\Bbb{Z}[i]/I = \{ a+bi : a,b \in \{ 0,1,2,3,4\} \}$, so it has $25$ elements.

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