A topological expression for dual-classical Krull dimension of rings

Sayed Malek Javdannezhad; Nasrin Shirali

Home/Archive/arXiv

arXiv · 2309.10541

A topological expression for dual-classical Krull dimension of rings

Sayed Malek Javdannezhad·Nasrin Shirali

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GN math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sayed Malek Javdannezhad, Nasrin Shirali. 2023-09-19. A topological expression for dual-classical Krull dimension of rings. https://arxiv.org/abs/2309.10541

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sobriety of Scott topologies under countability conditions

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

math.GN↗

arXiv2026

Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness

Lawson and Mislove posed the following problem in 1990 as Problem~535 in \emph{Open Problems in Topology}: for a core-compact space $X$ and a dcpo $P$ equipped with its Scott topology, under what conditions on $P$ do the Isbell and Scott topologies on $C(X,P)$ agree?It was proved that, for a nonempty bicomplete continuous dcpo $P$, the Isbell and Scott topologies on $C(X,P)$ coincide for every core-compact space $X$ if and only if $P$ is bounded complete; for every compact core-compact space $X$ if and only if $P$ is conditionally bounded complete; and for every RW-space $X$ if and only if $P$ is a pointed continuous $L$-domain.We remove the bicompleteness assumption from all three classifications by combining the forbidden-retract theorem of Jia, Jung and Li with a separation theorem for powers of downward well-ordered chains and suitable Alexandrov test spaces.

math.GN↗

arXiv2026

Three Problems on Separable Quotients of Precompact Abelian Groups

We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\T^{\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \cite{LMT}.

math.GN↗