Proof algebraic number fields Janusz, implies that every dedekind Ring is principal?

Question

The book is Algebraic number fields,

The ring $R$ is a Dedekind ring and $\mathcal{U}$ is a ideal of $R$. Janusz

The first four lemmas is completely clear to me, the While i can't understand the last corollary.

For example if $B$ is Dedekind Ring, then this implies that $R$ have a finite number of prime ideals is principal. Clearly is incorrect.

Is there an error in this corollary?

Thank you all.

Answer 1

Okay, maybe this is where you're confused: if $B$ were Dedekind, $(0)$ is already prime. Then Corollary 3.7 would say merely that $(0) = (0)$. The remaining work just serves to show that $B\cong B$, which is true but doesn't tell us anything about the non-zero prime ideals of $B$.