Document Type: Research Note
Inonu University, Science and Art Faculty Department of Mathematics, Malatya, Turkey
We prove that the set of homotopy classes of the paths in a topological ring is a topological ring
object (called topological ring-groupoid). Let p : X → X be a covering map and let X be a topological ring.
We define a category UTRCov(X) of coverings of X in which both X and X have universal coverings, and a
category UTRGdCov( π1X ) of coverings of topological ring-groupoid π1X , in which X and R0 = X have
universal coverings, and then prove the equivalence of these categories. We also prove that the topological
ring structure of a topological ring-groupoid lifts to a universal topological covering groupoid.