A topological expression for dual-classical Krull dimension of rings

TL;DR

This paper introduces a topological framework for analyzing the dual-classical Krull dimension of rings using the SH-topology on the set of non-zero strongly hollow ideals, establishing connections between topological properties and ring dimensions.

Contribution

It defines the SH-topology on the set of non-zero strongly hollow ideals and links topological properties with the dual-classical Krull dimension of rings, including the case of D-rings.

Findings

X is Noetherian iff every subset is quasi-compact

R has dcc on semi-sh-ideals iff X is Noetherian

If X has derived dimension, then R has dual-classical Krull dimension

Abstract

Let R be a ring and X = SH(R)-{0} be the set of all non-zero strongly hollow ideals (briefly, sh-ideals) of R. We first study the concept SH-topology and investigate some of the basic properties of a topological space with this topology. It is shown that if X is with SH-topology, then X is Noetherian if and only if every subset of X is quasi-compact if and only if R has dcc on semi-sh-ideals. Finally, the relation between the dual-classical Krull dimension of R and the derived dimension of X with a certain topology has been studied. It is proved that, if X has derived dimension, then R has the dual-classical krull dimension and in case R is a D-ring (i.e., the lattice of ideals of R is distributive), then the converse is true. Moreover these two dimension differ by at most 1.