Related papers: On Continuous and Adjoint Morphisms Between Noncom…

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…

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially ``sewn…

Given a ring R and S one of its proper ideals, we obtain a compactification of the prime spectrum of S through a mainly algebraic process. We name it the R-nilcompactification of SpecS. We study some categorical properties of this…

Let $R\rightarrow S$ be an arbitrary ring extension of Noetherian rings. In this article we study the behaviour of Zariski closedness of the support of Lyubeznik functors $\mathrm{T}$, when the ring extension $R\rightarrow S$ is namely…

We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism…

For a commutative ring $R$ with unit $1\ne 0$ and a multiplicatively closed subset $S$ of $R$, we introduce a new topology on the $S$-prime spectrum $\mathrm{Spec}_SR$ of $R$ called the $S$-flat topology. Our aims is to give an algebraic…

Given a flat local ring homomorphism R\to S, and two finitely generated R-modules M and N, we describe conditions under which the modules Tor^i(M,N) and Ext^i(M,N) have S-module structures that are compatible with their R-module structures.

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space $\scal(R)$ of prime semigroups of $R$. We show that, under…

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 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…

Let $R$ be a commutative local ring. We study the subcategory of the homotopy category of $R$-complexes consisting of the totally acyclic $R$-complexes. In particular, in the context where $Q\to R$ is a surjective local ring homomorphism…

In this work we define a primary spectrum of a commutative ring R with its Zariski topology $\mathfrak{T}$. We introduce several properties and examine some topological features of this concept. We also investigate differences between the…

For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X \rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of…

Let R be a commutative Noetherian ring, I and J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then…

We associate to any ring $R$ with identity a partially ordered set Hom$(R)$, whose elements are all pairs $(\mathfrak a,M)$, where $\mathfrak a=\ker\varphi$ and $M=\varphi^{-1}(U(S))$ for some ring morphism $\varphi$ of $R$ into an…

We uncover the very rich graph topology of generic bounded non-Hermitian spectra, distinct from the topology of conventional band invariants and complex spectral winding. The graph configuration of complex spectra are characterized by the…

Given a scheme S and a flat morphism T \to S of finite presentation we define a surjective S-morphism to an {\'e}tale and separated S-scheme, which is universal in an obvious sense. Properties of this morphism are deduced from a thorough…

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of…

Given a bounded complex of finitely generated modules $M$ over a commutative noetherian local ring $R$, one assigns to it a variety, $\mathcal V_R(M)$, called the cohomological support variety of $M$ over $R$. The variety $\mathcal V_R(M)$…