The dual-classical Krull dimension of rings via topology

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

2 Department of Science, Shahid Rajaee Teacher Training University: Tehran, Tehran, Iran

10.48308/cgasa.2025.238090.1531

Abstract

Let $R$ be a ring and  $\mathcal{X} = \mathcal{SH}(R)-\{0\}$ be the set   all  of 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  $\mathcal X $ is  with $SH$-topology, then  $\mathcal {X}$ is Noetherian if and only if every subset of $\mathcal X$ 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  $\mathcal {X}$ with a certain topology has been studied. It is proved that,  if $\mathcal {X}$ 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 $1$. 

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References

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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 24 September 2025

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