학술논문

Some aspects of Zariski topology for multiplication modules and their attached frames and quantales
Document Type
Article
Source
대한수학회지, 56(5), pp.1285-1307 Sep, 2019
Subject
수학
Language
English
ISSN
2234-3008
0304-9914
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 $M$.