Unramified class field theory for orders

The main theorem of unramified class field theory, which states that the class group of the ring of integers of a number field $K$, is canonically isomorphic to the Galois group of the maximal totally unramified abelian extension of $K$ over $K$, is generalized and proved for all infinite commutative rings with unit that, like rings of integers, are connected and finitely generated as a module over ${\mathbf{Z}}$. Modulo their nilradical, these rings are exactly the connected orders in products of number fields.