find all the prime ideals of $\mathbb{Z}_8$

Asked May 22 '22 at 02:16

Active May 22 '22 at 11:38

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find all the prime ideal of $\mathbb{Z}_8$?

My attempt : Positive divisor of $8$ are $1, 2, 4$ and $8$

So the ideal in $\mathbb{Z}_8$ are

$(1)=\mathbb{Z}_8$

$(2)=\{0,2,4,6\}$

$(4)=\{0,4\}$

$(8)=\{0\} $

Here $(4)$ is not prime ideal because $2.2 \in 4$ but $2\notin (4 ) $

Therefore the prime ideals are $(1) ,(2)$ and $(8)$ because $1.1 \in (1) $ , $2.2 \in (2)$ and $0.0\in \{0\}$

asked May 22 '22 at 02:16

wasiu

1

$2\cdot 4 \in {0}$ but $2\notin {0}$ and $4\notin{0}$ – jjagmath May 22 '22 at 02:31
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Also, the whole ring is not prime. It's explicitly excluded in the definition of prime ideal. – jjagmath May 22 '22 at 02:32
okay @jjagmath Good logic – wasiu May 22 '22 at 02:35
A more general duplicate. – Dietrich Burde May 22 '22 at 08:19

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