Rings with a discrete group of divisor classes

We shall say that a ring $A$ has a DGC (discrete group of classes) if the group of divisor classes is preserved in going to the ring of formal power series, i.e. $C(A)\to C(A[[T]])$ is an isomorphism. We prove the localness and faithfully flat descent of the DGC property. We establish a connection between the DGC property of a ring and its depth. We also give a characterization of two-dimensional rings with DGC and characteristic zero rings with DGC. Finally, it is shown that the discreteness of the group of divisor classes is preserved under regular extensions of rings such as $A[T_1,\dots,T_n]$, $A[[T_1,\dots,T_n]]$, completions, etc.
Bibliography: 13 titles.