Dedekind not implies PID
From Commalg
This article gives the statement and possibly, proof, of a nonimplication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property need not satisfy the second commutative unital ring property
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Statement
There exist Dedekind domains which are not principal ideal domains.
Proof
In fact, any ring of integers in a number field is a Dedekind domain, but most of them are not PIDs (in fact, they are not even unique factorization domains). For instance, consider the ring: