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 ?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
?X are equivalent. Recently it was proved that if (X, x0) is an H-group,
then the fundamental groupoid ?X is a categorical group and the category of
the covering spaces of (X, x0) is equivalent to the category of covering
groupoids of the categorical group ?X. The purpose of this paper is to
present similar results for rings and categorical rings.