# Alexandroff topology of algebras over an integral domain

`2019-12-04 10:30:00``2019-12-04 11:30:00``Alexandroff topology of algebras over an integral domain``Let S be an integral domain with field of fractions F, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R lies over S and the localization of R with respect to S\{0} is A. Let X be the set of all S-nice subalgebras of A. We define a notion of open sets on X which makes this set a T_0-Alexandroff space. This enables us to study the algebraic structure of X from a topological point of view. We prove that an irreducible subset of X has a supremum with respect to the specialization order. We present equivalent conditions for an open set of X to be irreducible and characterize the irreducible components of X. We also characterize quasi-compactness of subsets of a T_0-Alexandroff topological space.``Third floor seminar room (room 201, building 216)``אוניברסיטת בר-אילן - Department of Mathematics``mathoffice@math.biu.ac.il``Asia/Jerusalem``public`Let S be an integral domain with field of fractions F, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R lies over S and the localization of R with respect to S\{0} is A. Let X be the set of all S-nice subalgebras of A. We define a notion of open sets on X which makes this set a T_0-Alexandroff space. This enables us to study the algebraic structure of X from a topological point of view. We prove that an irreducible subset of X has a supremum with respect to the specialization order. We present equivalent conditions for an open set of X to be irreducible and characterize the irreducible components of X. We also characterize quasi-compactness of subsets of a T_0-Alexandroff topological space.

Last Updated Date : 06/11/2019