A \emph{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 \emph{group-groupoid} or 2-\emph{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.