Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 25 1 2025 09 24 The dual-classical Krull dimension of rings via topology 106251 10.48308/cgasa.2025.238090.1531 EN Nasrin Shirali Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran 0000-0002-9907-7352 Sayed Malek Javdannezhad Department of Science, Shahid Rajaee Teacher Training University: Tehran, Tehran, Iran Journal Article 2024 12 27 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 $1$. https://cgasa.sbu.ac.ir/article_106251_b2e1a832a852780f31d0640a908b6af9.pdf