A categorical group is a kind of categorization of group and similarly a categorical ring is a categorization of ring. For a topological group X, the fundamental groupoid pi X is a group object in the category of groupoids, which is also called in literature group-groupoid or 2-group. If X is a path connected topological group which has a simply connected cover, then the category of covering groups of X and the category of covering groupoids of pi X are equivalent. Recently it was proved that if (X, x(0)) is an H-group, then the fundamental groupoid pi X is a categorical group and the category of the covering spaces of (X, x(0)) is equivalent to the category of covering groupoids of the categorical group pi X. The purpose of this paper is to present similar results for rings and categorical rings.