Open AccessSome topology on zero-dimensional subrings of product of rings
Author(s) -
Hassan Mouadi,
Driss Karim
Publication year - 2020
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil2014589m
Subject(s) - mathematics , topology (electrical circuits) , zero (linguistics) , ultrafilter , direct limit , limit (mathematics) , product topology , ring (chemistry) , product (mathematics) , general topology , point (geometry) , pure mathematics , discrete mathematics , combinatorics , topological space , mathematical analysis , geometry , philosophy , linguistics , chemistry , organic chemistry
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.