A domain $D$ is integrally closed if it is integrally closed in its field of fractions $F$. A Noetherian ring may not be integrally closed; for example, $\mathbf{Z}\sqrt{5}$ is Noetherian but it is not integrally closed in its field of fractions $\mathbf{Q}\sqrt{5}$.
It is said that every Dedekind domain is integrally closed. Does the converse hold, namely are integrally closed domains Dedekind Domains?
The only example of an integrally closed domain that I have is the ring of integers and the examples of a Dedekind domain are PID and the ring of integers of a number field. May you please provide an example of an integrally closed domain which is not Dedekind?
Would you help me, please? Thank you in advance.
No. Among many characterisations, Dedekind domains are noetherian integrally closed domains of Krull dimension $1$, i.e. every non-zero prime ideal is maximal.
As a simple counter example, if $K$ is a field, the ring of polynomials $K[X_1,\dots, X_n]$ is a noetherian integral domain of Krull dimension $n$.
Furthermore, a valuation domain of dimension (height) $1$ is integrally closed (as all valuation domains), but it is not necessarily a Dedekind domain, as it is not necessarily a discrete valuation domain.