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Let      Z Z       denote the integers,      Q Q       the rationals,      X X       an indeterminate and      G G       a finitely generated abelian group. Then there is a Dedekind domain      D D       such that      Z [ X ] ⊂  D ⫋  Q [ X ] Z[X] \subset D \varsubsetneqq Q[X]      , and      D D       has class group      G G      . If      G = 0 G = 0       then      D D       is a noneuclidian PID.

More noneuclidian 𝑃𝐼𝐷’s and Dedekind domains with prescribed class group