zariski topology

Let G be a group, R be a G-graded commutative ring with identity, M be a unitary graded R-module, Specg(R) be the set of graded prime ideals of R, and Cl.Specg(M) be the set of all graded classical prime submodules of M. In this paper among other things, the author studied the Zariski topology on both and Cl.Specg(M), and investigate some properties of the Zariski topology on Cl.Specg(M) and some conditions under which the graded classical prime spectrum of M is a spectral for its Zariski topology.

Residuated lattices are the major algebraic counterpart of logics without contraction rule, as they are more generalized logic systems including important classes of algebras such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and De Morgan residuated lattices among others, on which filters and ideals are sets of provable formulas. This paper presents a meaningful exploration of the topological properties of prime ideals of residuated lattices. Our primary objective is to endow the set of prime ideals with the stable topology, a topological framework that proves to be more refined than the well-known Zariski topology. To achieve this, we introduce and investigate the concept of pure ideals in the general framework of residuated lattices. These pure ideals are intimately connected to the notion of annihilator in residuated lattices, representing precisely the pure elements of quantales. In addition, we establish a relation between pure ideals and pure filters within a residuated lattice, even though these concepts are not dual notions. Furthermore, thanks to the concept of pure ideals, we provide a rigorous description of the open sets within the stable topology. We introduce the i-local residuated lattices along with their properties, demonstrating that they coincide with local residuated lattices. The findings presented in this study represent an extension beyond previous work conducted in the framework of lattices, and classes of residuated lattices.

Following [6], we define Grothendieck topologies on a small category and describe sheaves for these Grothendieck topologies. This generalizes, in a natural way, the theory of sheaves on a topological space.

In this paper, we present some characterizations of prime and maximal filters. Moreover, we introduce $\cap$-irreducible filters of an equality algebra, investigate some results about them and relations between maximal, prime, $\vee$-irreducible and $\cap$-irreducible filters in equality algebra. Also, we introduce spectrum of an equality algebra and prove that the spectrum endowed with Zariski topology is a compact T_0 topological space and maximal spectrum (as a subspace of that) is a compact T_1 topological space.

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 topological space.

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. Let $FSpec(M)$ denotes the collection of all prime fuzzy submodules of $M$. In this regards some basic properties of Zariski topology on $FSpec(M)$ are investigated. In particular, we prove some equivalent conditions for irreducible subsets of this topological space and it is shown under certain conditions $FSpec(M)$ is a $T_0-$space or Hausdorff.

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