on two problems concerning the zariski topology of modules

On two problems concerning the Zariski topology of modules

Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...

Zariski-Like Topology on the Classical Prime Spectrum of a Modules

PRIMARY ZARISKI TOPOLOGY ON THE PRIMARY SPECTRUM OF A MODULE

‎‎Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎We define the primary spectrum of $M$‎, ‎denoted by $mathcal{PS}(M)$‎, ‎to be the set of all primary submodules $Q$ of $M$ such that $(operatorname{rad}Q:M)=sqrt{(Q:M)}$‎. ‎In this paper‎, ‎we topologize $mathcal{PS}(M)$ with a topology having the Zariski topology on the prime spectrum $operatorname{Spec}(M)$ as a sub...

zariski-like topology on the classical prime spectrum of a modules

The Basic Zariski Topology

We present the Zariski spectrum as an inductively generated basic topology à la Martin-Löf and Sambin. Since we can thus get by without considering powers and radicals, this simplifies the presentation as a formal topology initiated by Sigstam. Our treatment includes closed and open subspaces: that is, quotients and localisations. All the effective objects under consideration are introduced by ...

on direct sums of baer modules

the notion of baer modules was defined recently