Ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$

Question

I need to describe all the ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ I suppose that trivials, and $(0,..,1_{i},..,0)\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ for any $i$ and nilradicals in all components

How to show that they r unique(if thats true)

P.S. I mean $(0,..,1_{i},..,0)$ for chinese remainder theorem decomposition(pardon my eng)

Answer 1

Suppose $R$ is a ring and $I$ an ideal; then the ideals of $R/I$ are $$\frac{J}{I}$$ with $J \subseteq R$ ideal containing $I$.

In your case $I=(p_{1}^{k_{1}} \ldots p_{m}^{k_{m}})$; $\Bbb{Z}$ is a PID so $ J=(a)$ and $$I \subseteq J \Longleftrightarrow a \mid p_{1}^{k_{1}} \ldots p_{m}^{k_{m}}$$ So the ideal of $R/I$ are $$(p_{1}^{j_{1}} \ldots p_{m}^{j_{m}})/(p_{1}^{k_{1}} \ldots p_{m}^{k_{m}})$$ with $j_r \leq k_r $ for all $r$.