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

let $r$ be a commutative ring with identity and $m$ be an$r$-module. let $fspec(m)$ denotes the collection of all prime fuzzysubmodules of $m$. in this regards some basic properties of zariskitopology on $fspec(m)$ are investigated. in particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$fspec(m)$ is a $t_0-$spa...

Let f(z) = z +az + bz + cz+ d ∈ Z[z] and let us consider a del Pezzo surface of degree one given by the equation Ef : x 2 − y − f(z) = 0. In this note we prove that if the set of rational points on the curve Ea, b : Y 2 = X + 135(2a − 15)X − 1350(5a + 2b − 26) is infinite, then the set of rational points on the surface Ef is dense in the Zariski topology.

Zariski groups are @0-stable groups with an axiomatically given Zariski topology and thus abstract generalizations of algebraic groups. A large part of algebraic geometry can be developed for Zariski groups. As a main result, any simple smooth Zariski group interprets an algebraically closed eld, hence is almost an algebraic group over an algebraically closed eld.

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...

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...

The nature of finitely generated infinite index subgroups of SL(3,Z) remains extremely mysterious. It follows from the famous theorem of Tits [12] that free groups abound and, moreover, Zariski dense free groups abound. Less trivially, classical arithmetic considerations (see for example §6.1 of [9]) can be used to construct surface subgroups of SL(3,Z) of every genus ≥ 2. However these are con...