Van der Waerden rings

The study of the class of van der Waerden rings was initiated by Comfort, Remus and Szambien in [7]. A compact ring (R,T) is a van der Waerden ring if and only if T is the only totally bounded ring topology on R. In this paper the study of this class of compact rings is continued. We prove that an Abelian compact van der Waerden ring R has the form R=(∏i∈IRi)×L, where L is a compact radical ring with radical topology and Ri(i∈I) is a compact local topologically finitely generated ring. It is shown that every van der Waerden ring is totally disconnected and metrizable. Furthermore, the class of van der Waerden rings is closed under extensions. Different classes of compact rings are studied which are closely related to van der Waerden rings.