Some aspects of Zariski topology for multiplication modules and their attached frames and quantales
J. Korean Math. Soc. 2019 Vol. 56, No. 5, 1285-1307
https://doi.org/10.4134/JKMS.j180649
Published online September 1, 2019
Jaime Castro, Jos\'{e} R\'{\i}os, Gustavo Tapia
Instituto Tecnol\'{o}gico y de Estudios Superiores de Monterrey; Universidad Nacional Aut\'{o}noma de M\'{e}xico; Universidad Aut\'{o}noma de Ciudad Ju\'{a}rez
Abstract : For a multiplication $R$-module $M$ we consider the Zariski topology in the set $Spec\left( M\right) $ of prime submodules of $M$. We investigate the relationship between the algebraic properties of the submodules of $M$ and the topological properties of some subspaces of $Spec\left( M\right) $. We also consider some topological aspects of certain frames. We prove that if $ R $ is a commutative ring and $M$ is a multiplication $R$-module, then the lattice $Semp\left( M/N\right) $ of semiprime submodules of $M/N$ is a spatial frame for every submodule $N$ of $M$. When $M$ is a quasi projective module, we obtain that the interval $\mathcal{\uparrow } (N)^{Semp\left( M\right) }=\left\{ P\in Semp\left( M\right) \mid N\subseteq P\right\} $ and the lattice $Semp\left( M/N\right) $ are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of $M$.
Keywords : Zariski topology, multiplication modules, frames, spatial frames, quantales, dense subspaces, irreducible subspaces, Krull dimension.
MSC numbers : 16S90, 16D50, 16P50, 16P70
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