Let R be a ring and {Ri}i?I a family of zero-dimensional rings. We define the
Zariski topology on Z(R,?Ri) and study their basic properties. Moreover,
we define a topology on Z(R,?Ri) by using ultrafilters; it is called the
ultrafilter topology and we demonstrate that this topology is finer than the
Zariski topology. We show that the ultrafilter limit point of a collections
of subrings of Z(R,?Ri) is a zero-dimensional ring. Its relationship with
F-lim and the direct limit of a family of rings are studied.
Let R be a subring of a ring T, and let F be a non-principal ultrafilter on the natural numbers IN. We consider properties and applications of a countably compact, Hausdorff topology called the "F-topology" defined on space of all zero-dimensional subring of T that contains a fixed subring R. We show that the F-topology is strictly finer than the Zariski topology. We extend results regarding distinguished spectral topologies on the space of zero-dimensional subring.
An extension of rings R T is said to be a zero-dimensional pair if all intermediate rings between R and T are zerodimensional. In this paper, we investigate the question of when a pair Q aPA R a ;Q aPA T a is zero-dimensional, where each R a ; T a is a zero-dimensional pair. Our main result gives an answer in the case R a and T a are quasilocal for each a.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.