Related papers: Relative Zariski Open Objects

Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on…

In the preprint arXiv:2511.07900 we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative…

Here we continue to characterize a recently introduced notion, le-modules $_{R}M$ over a commutative ring $R$ with unity \cite{Bhuniya}. This article introduces and characterizes Zariski topology on the set $Spec(M)$ of all prime submodule…

In arXiv:2511.04191 we constructed schemes of objects in small categories which contained a set of basepoints with local representing (localizing) objects. Here we prove that the category $\cat{Rings}$ of associative rings with unit has a…

Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the…

A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of…

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings.…

The purpose of this paper is to introduce a Zariski-like topology on the spectrum of all proper ideals of a ring. We show that the space is T_0, quasi-compact, and every irreducible closed subset has a unique generic point. Furthermore,…

We introduce a dual Zariski topology on the spectrum of fully coprime $R$-submodules of a given duo module $M$ over an associative (not necessarily commutative) ring $R$. This topology is defined in a way dual to that of defining the…

In this paper we introduce and investigate top (bi)comodules} of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that…

Over the past two decades several different approaches to defining a geometry over ${\mathbb F}_1$ have been proposed. In this paper, relying on To\"en and Vaqui\'e's formalism, we investigate a new category…

We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local - i.e. it can be verified on any open affine covering of the scheme - and it agrees, in their setting, with the notion studied by Murfet…

We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…

The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to strongly irreducible ideals (endowed with Zariski topologies) of…

Let $R$ be a commutative ring with nonzero identity and $M$ be an $R$-module. Quasi-prime submodules of $M$ and the developed Zariski topology on $q\Spec(M)$ are introduced. We also, investigate the relationship between the algebraic…

Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be the space of valuation rings of $F$ that dominate $D$. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of…

Let M be a module over a commutative ring and let Spec(M) (resp. Max(M)) be the collection of all prime (resp. maximal) submodules of M. We topologize Spec(M) with Zariski topology, which is analogous to that for Spec(R), and consider…

The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the…

We construct the canonical structure of an irreducible projective variety on the set of connected curves of degree $d$ in $\Bbb P^n$ with rational components (some components can be multiple). The set of rational curves is open subset in…

This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over $\Z$ that admits a torification. Toric varieties,…