Suppose $p$ is a prime factor of $n$. We have $\mathbb{Z}/n\mathbb{Z}$ consisting of all integers mod $n$ and $p\mathbb{Z}/n\mathbb{Z}$ consisting of all multiples of $p$ mod $n$, a prime ideal.
Is this exactly the same as $(p)$, just with different notation? Do the notations have different "meanings" in any way?
This is a very simple question, but I haven't found this laid out explicitly anywhere.