# Example of non-Krull integrally closed BFD?

Here's another question in the same spirit as my previous one:

Are there any integrally closed BFDs which are not Krull domains?

Some background information:

A BFD (bounded factorization domain) is defined as an atomic domain where the number of factors is bounded for each element. (More precisely: every nonzero nonunit $x$ can be factorized into irreducibles, but not necessarily uniquely, and there is a number $N(x)$ such that whenever $x=a_1 \dots a_r$ is such a factorization, then $r \le N(x)$.)

Unless I've misread something, the following relations hold (with strict inclusions): $$\{ \text{Noetherian domains} \} \subset \{ \text{BFDs} \} \quad \Bigl( \subset \{ \text{ACCP domains} \} \Bigr),$$ $$\{ \text{Noetherian domains} \} \cap \{ \text{integrally closed domains} \} = \{ \text{Noetherian domains} \} \cap \{ \text{Krull domains} \},$$ $$\{ \text{Krull domains} \} \subset \{ \text{integrally closed domains} \},$$ $$\{ \text{Krull domains} \} \subset \{ \text{BFDs} \}.$$ (Disclaimer: Trying to navigate in this jungle of classes of integral domains has got my head spinning, so I'm not 100% sure about anything at the moment...)

For the purposes of drawing a correct Venn diagram involving these relations (and many more!), I would need to know whether there is equality or strict inclusion in the relation $$\{ \text{Krull domains} \} \subseteq \{ \text{BFDs} \} \cap \{ \text{integrally closed domains} \}.$$

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