Let Rbe a Dedekind domain with quotient field Kand let be the integral closure of Rin an algebraic closure of K

Let Γ be the set of rings Asuch that Ais finitely generated over R and R ⊆ A ⊆ .Let  be the set of rings O such that O is finitely generated over  .We introduce an equivalence relation on   and prove that the properties of a ring A ε Γ can be explicitly derived from the properties of an element equivalent to   and monogenic. This allows to generalize Dedekind's criterion, to describe prime ideals, the P-radical and primary decomposition in Aas well as to generalize Kummer's factorization theorem to all Dedekind domains.