I have just shown that $\mathbb{Z}_{n}$ is a principal ideal ring, by using (1) a previously proven result that $\mathbb{Z}$ is a principal ideal domain (and hence a principal ideal ring), and (2) the result that the image of a principal ideal ring under a ring endomorphism (in this case, the canonical epimorphism $f: \mathbb{Z} \to \mathbb{Z}_{n} \simeq \mathbb{Z}/n\mathbb{Z}$ defined by $a \mapsto a + n\mathbb{Z} \simeq m\mathbb{Z}, \, m \in \mathbb{Z}_{n}$) is also a principal ideal ring.
By this result (or rather as a consequence of the proof of this result), I determined that ideals of $\mathbb{Z}_{n}$ are of the form $f(n\mathbb{Z})$, so does that mean that all the ideals are of the form $m\mathbb{Z}$ as described above in my mention of the canonical epimorphism?
Now, by the First Isomorphism Theorem for Rings, Extended Version, the possible homomorphic images of $\mathbb{Z}_{n}$ are isomorphic to $\mathbb{Z}_{n}/I$, where $I$ is an ideal of $\mathbb{Z}_{n}$. So, once I know how to correctly write down the ideals of $\mathbb{Z}_{n}$, I should be able to describe completely all homomorphic images of $\mathbb{Z}_{n}$, which is what I am looking to do here.
So I suppose my question here is twofold: (1) Have I correctly identified/described the ideals of $\mathbb{Z}_{n}$ here? If not, what should I change? And (2) Once I have correctly done that, is it enough to just say that all homomorphic images are $\mathbb{Z}_{n}/I$ where the $I$ are those ideals, or is there a better way to express them?
Thank you in advance.
$\newcommand{\Z}{\mathbb Z}$ By the Third Isomorphism Theorem, ideals $I$ of $\Z/n\Z$ are in bijection of ideals $J$ of $\Z$ that contain $(n)$, and furthermore, $(\Z/n\Z)/I \cong \Z/J$ under this correspondence. So now you just need to figure out what ideals of $\Z$ contain $(n)$.