powers of prime ideals in a Dedekind domain

Question

Why in a Dedekind domain, $p^r \neq p^{r + 1}$ for any prime ideal $p$ and integer $r$?

Answer 1

A characterization of Dedekind domains is that non-zero ideals are invertible.

If $\mathfrak p^r = \mathfrak p^{r+1}$ then multiplying both sides by $\mathfrak p^{-r}$ gives $1 = \mathfrak p$.