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The goal is to find all ideals of $\mathbb{Z}_n$.

First of all, I would like to know wheter this is a set of all congruence classes modulo $n$ or a set stricted to $\{0, 1, ..., n-1 \}$ with modular arithmetics? Because I do not know which one is easier to work with.

Secondly, besides trivial ideals i.e. $\{0\}, \mathbb{Z}_n$ how do I find others? I do know that it will be somehow connected to the primes dividing $n$ and such ideals will be maximal, but what with other integers being composite of prime divisors of $n$?

Would be grateful for any hints to start with because I know something in theory, but I have trouble with writing this down.

janusz
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  • This might be helpful: https://math.stackexchange.com/questions/2483021/what-are-the-ideals-of-mathbbz-n-mathbbz – fish May 13 '20 at 13:09

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Hint: ideals of $\mathbb{Z}_n$ are in correspondence with ideals of $\mathbb{Z}$ that contain $n\mathbb{Z}$. Characterize the subgroups of the integers, prove that they are all ideals. Now pick the ones that contain $n\mathbb{Z}$ and project them into the quotient $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$.

qualcuno
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  • Ok, so I do know that all additive subgroups of $\mathbb{Z}$ are sets of form $(d) = {k \cdot d | k \in \mathbb{Z} }$ where $d \in \mathbb{Z}$. Showing that this is indeed ideal is also not that hard since $\forall (kd) \in (d)$ and $\forall n \in \mathbb{Z}$ we have $(kd)n = (kn)d \in (d)$. My question is how do I pick the one that contains $n\mathbb{Z}$? – janusz May 13 '20 at 13:24