Can $\Bbb Z_n$l ideals be generated by one element?

Question

Let's define the ring, $\Bbb Z_n$ (It is surely a well-known ring.)

1. set : $Z_n$ ={$[0]_{n}$, $[1]_{n}$, ... $[n-1]_{n} \}$

2. operation : addition or multiplication for the $mod n$

Say the $m \in \Bbb Z_n$, It is obvious that $\langle m \rangle$ is an ideal of $\Bbb Z_n$.

Then, $\forall$ ideal J of the $Z_n$, can the ideal J be expressed as the $\langle m \rangle$ ?

Answer 1

consider $I\subset \mathbb{Z}_n$ be ideal.Let $\eta:\mathbb{Z}\to \mathbb{Z}_n$ be natural projection then observe $\eta^{-1}(I)$ is ideal of $\mathbb{Z}$. So it is generated by a single element (say a) as $\mathbb{Z}$ being PID. Now it is easy to see $I$ is generated by $\eta(a)$.

Answer 2

Since $\mathbb{Z}_n\cong\frac{\mathbb{Z}}{\langle n\rangle}$ and $\mathbb{Z}$ is a PID, we have $\mathbb{Z}_n$ is also a PID.