let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

Let $R$ be a commutative ring with identity and $M$ be a unitary$R$-module. The primary-like spectrum $Spec_L(M)$ is thecollection of all primary-like submodules $Q$ such that $M/Q$ is aprimeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ isa prime submodule for all $Qin Spec_L(M)$. This class containsthe family of multiplication modules properly. The purpose of thispaper is to intro...

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...

Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...

$r$-module. in this paper, we explore more properties of $max$-injective modules and we study some conditions under which the maximal spectrum of $m$ is a $max$-spectral space for its zariski topology.

in this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.

$R$-module. In this paper, we explore more properties of $Max$-injective modules and we study some conditions under which the maximal spectrum of $M$ is a $Max$-spectral space for its Zariski topology.

The starting point for this dissertation is whether the concept of Zariski geometry, introduced by Hrushovski and Zilber, could be generalized to the context of nonelementary classes. This leads to the axiomatization of Zariski-like structures. As our main result, we prove that if the canonical pregeometry of a Zariski-like structure is non locally modular, then the structure interprets either ...