Zariski Topology Research Articles

Zariski topology on the spectrum of intuitionistic fuzzy classical primary submodules

Zariski topology on the spectrum of intuitionistic fuzzy classical primary submodules

Graded pseudo weakly prime spectrum of graded topological modules

In this study, we introduce graded pseudo weakly prime submodules of G-graded R-modules, which are an extension of graded weakly prime ideals over G-graded rings. On the graded spectrum of graded pseudo weakly prime submodules, we investigate the Zariski topology. Different aspects of this topological space are investigated, and they are linked to the algebraic properties of the G-graded R-modules under study.

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be a homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case and which comes from the theory of real closed fields otherwise.

The d-elements of precoherent preidempotent quantales and their applications

Abstract In this paper, we introduce the notion of d-elements on precoherent preidempotent quantale (PIQ), construct Zariski topology on $Max(Q_{d})$ and explore its various properties. Firstly, we give a sufficient condition of a topological space $Max(Q_{d})$ being Hausdorff. Secondly, we prove that if $ P=\mathfrak{B}(P) $ and $ Q=\mathfrak{B}(Q) $, then $P$ is isomorphic to $Q$ iff $ Max(P_{d}) $ is homeomorphic to $ Max(Q_{d}) $. Moreover, we prove that $ (P\otimes Q)_{d} $ is isomorphic to $ P_{d} \otimes Q_{d} $ iff $ P_{d} \otimes Q_{d}=(P_{d} \otimes Q_{d})_{d} $. Finally, we prove that the category $ \textbf{dPFrm} $ is a reflective subcategory of $\textbf{PIQuant}.$

On simple evolution algebras of dimension two and three. Constructing simple and semisimple evolution algebras

This work classifies three-dimensional simple evolution algebras over arbitrary fields. For this purpose, we use tools such as the associated directed graph, the moduli set, inductive limit group, Zariski topology and the dimension of the diagonal subspace. Explicitly, in the three-dimensional case, we construct some models i III λ 1 , … , λ n p , q of such algebras with 1 ≤ i ≤ 4 , λ i ∈ K × , p , q ∈ N , such that any algebra is isomorphic to one (and only one) of the given in the models and we further investigate the isomorphic question within each one. Moreover, we show how to construct simple evolution algebras of higher-order from known simple evolution algebras of smaller size.

Avoidance Spectrum of Alexandroff Spaces

In this paper we prove that every T0 Alexandroff topological space (X, τ ) is homeomorphic to the avoidance of a subspace of (Spec(Λ), τZ), where Spec(Λ) denotes the prime spectrum of a semi-ring Λ induced by τ and τZ is the Zariski topology. We also prove that (Spec(Λ), τZ) is an Alexandroff space if and only if Λ satisfies the Gilmer property.

Zariski topology on the secondary-like spectrum of a module

Abstract Let ℜ \Re be a commutative ring with unity and ℑ \Im be a left ℜ \Re -module. We define the secondary-like spectrum of ℑ \Im to be the set of all secondary submodules K K of ℑ \Im such that the annihilator of the socle of K K is the radical of the annihilator of K K , and we denote it by Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) . In this study, we introduce a topology on Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Spec s ( ℑ ) {{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.

A note on the Zariski topology on groups and semigroups

We show that the semigroup Zariski topology on a group can be strictly coarser than the group Zariski topology on it, answering a question from [7].

A note on the image of polynomials on upper triangular matrix algebras

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p ( x 1 , … , x m ) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p ( T n ( K ) ) is a dense subset of T n ( K ) ( r − 1 ) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023.

Some topological properties of the Zariski topology on $$PL.Spec_{g}(\Im )$$

Some topological properties of the Zariski topology on $$PL.Spec_{g}(\Im )$$

A STUDY ON GRAPHS DEFINED ON L-SLICES

Let L be a locale with top element 1L and J be a join semilattice with bottom element 0J. The L-slice (σ,J) is the action of the locale on join semilattice satisfying certain properties. The concept of L-slices were modelled in tune with the modules in algebra. The benefit of studying L-slices is that we can approach the structure algebraically as well as topologically. This paper deals with the graph theoretic approach to L-slices. The idea of relating graphs with algebraic structures was started by the work of Beck in [3]. The algebraic properties of L-slices prompted us to consider the possibility of various graphs that could be associated with L-slices. The article introduces two different graphs on L-slices. The total graph Γ((T(σ,J)) is defined. We derive a characterisation for such graphs to be nonempty. The structural properties of Γ((T(σ,J)) is studied. The weak Zariski Topology on (σ,J) gives us the graph GT(ω∗). The conditions under which the graph is nonempty is examined. Also some of the structural properties of GT(ω∗) is obtained.

Zip rings (resp. czip rings) and some applications

A commutative ring with identity is called a zip ring if each of its faithful ideals contains a finitely generated faithful ideal. A natural generalization of this notion is to require every faithful ideal to contain a countably generated faithful ideal. This is what we do in this paper. If a ring has this property, we call it a czip ring. Clearly, every zip ring is a czip ring. We give several characterizations of czip rings, including some in terms of subspaces of the space of minimal prime ideals with the Zariski topology. In rings of continuous functions we characterize the Tychonoff spaces X for which C(X) is a czip ring. This enables us to show that the class of czip rings properly contains that of zip rings.

On Zariski spaces of free acts over semigroups

This study is concerned with the collections of prime and radical subacts of free acts over a semigroup [Formula: see text] The collection of all prime subacts of any free act is topologized with two topologies as generalizations of the notion of the Zariski topology in rings and semigroups. Moreover, the interplay between some algebraic properties and some topological properties of the spectrum of free acts is investigated. In particular, the conditions under which the set of all prime subacts of free acts is a spectral space are studied. Also, it is shown that a certain type of Zariski space of free acts is isomorphic to a lattice of radical subacts.

Groups with cofinite Zariski topology and potential density

Tkachenko and Yaschenko [33] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is either almost torsion-free or of prime exponent). Furthermore, the authors connected this fact to Markov's notion of potential density and the existence of pairs of independent group topologies. Inspired by their work, we examine the class C of groups having cofinite Zariski topology in the general case, obtaining a number of very strong restrictions on these groups in the non-abelian case which suggest the bold conjecture that a group with cofinite Zariski topology is necessarily either abelian or finite. We show that Tkachenko–Yaschenko theorem fails in the non-abelian case and we offer a natural counterpart in the general case using a partial Zariski topology and an appropriate stronger version of the property almost torsion-free.

Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones

In this paper we explore the extent to which the algebraic structure of a monoid M M determines the topologies on M M that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or T 1 T_1 topologies; Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids and inverse monoids. If M M is a topological monoid such that every homomorphism from M M to a second countable topological monoid N N is continuous, then we say that M M has automatic continuity. We show that many well-known, and extensively studied, monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid N N \mathbb {N} ^\mathbb {N} ; the full binary relation monoid B N B_{\mathbb {N}} ; the partial transformation monoid P N P_{\mathbb {N}} ; the symmetric inverse monoid I N I_{\mathbb {N}} ; the monoid Inj ⁡ ( N ) \operatorname {Inj}(\mathbb {N}) consisting of the injective transformations of N \mathbb {N} ; and the monoid C ( 2 N ) C(2^{\mathbb {N}}) of continuous functions on the Cantor set 2 N 2^{\mathbb {N}} . The monoid N N \mathbb {N} ^\mathbb {N} can be equipped with the product topology, where the natural numbers N \mathbb {N} have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on N N \mathbb {N} ^\mathbb {N} , and its analogue on P N P_{\mathbb {N}} , is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on C ( 2 N ) C(2 ^\mathbb {N}) , and on the monoid C ( [ 0 , 1 ] N ) C([0, 1] ^\mathbb {N}) of continuous functions on the Hilbert cube [ 0 , 1 ] N [0, 1] ^\mathbb {N} . The symmetric inverse monoid I N I_{\mathbb {N}} has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid B N B_{\mathbb {N}} has no Polish semigroup topologies, nor do the partition monoids. At the other extreme, Inj ⁡ ( N ) \operatorname {Inj}(\mathbb {N}) and the monoid Surj ⁡ ( N ) \operatorname {Surj}(\mathbb {N}) of all surjective transformations of N \mathbb {N} each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on N N \mathbb {N} ^\mathbb {N} , P N P_{\mathbb {N}} , and Inj ⁡ ( N ) \operatorname {Inj}(\mathbb {N}) coincide with the pointwise topology; and we characterise the Zariski topology on B N B_{\mathbb {N}} . Along the way we provide many additional results relating to the Markov topology, the small index property for monoids, and topological embeddings of semigroups in N N \mathbb {N}^{\mathbb {N}} and inverse monoids in I N I_{\mathbb {N}} . Finally, the techniques developed in this paper to prove the results about monoids are applied to function clones. In particular, we show that: the full function clone has a unique Polish topology; the Horn clone, the polymorphism clones of the Cantor set and the countably infinite atomless Boolean algebra all have automatic continuity with respect to second countable function clone topologies.

The étale-open topology and the stable fields conjecture

For an arbitrary field K and a K -variety V , we introduce the étale-open topology on the set V(K) of K -points of V . This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when K is separably closed, real closed, or p -adically closed, respectively. Topological properties of the étale-open topology correspond to algebraic properties of K . For example, the étale-open topology on \mathbb{A}^1(K) is not discrete if and only if K is large. As an application, we show that a large stable field is separably closed.

The scheme of monogenic generators II: local monogenicity and twists

This is the second paper in a series of two studying monogenicity of number rings from a moduli-theoretic perspective. By the results of the first paper in this series, a choice of a generator $$\theta $$ for an A-algebra B is a point of the scheme $${\mathcal {M}}_{B/A}$$ . In this paper, we study and relate several notions of local monogenicity that emerge from this perspective. We first consider the conditions under which the extension B/A admits monogenerators locally in the Zariski and finer topologies, recovering a theorem of Pleasants as a special case. We next consider the case in which B/A is étale, where the local structure of étale maps allows us to construct a universal monogenicity space and relate it to an unordered configuration space. Finally, we consider when B/A admits local monogenerators that differ only by the action of some group (usually $${\mathbb {G}}_m$$ or $$\textrm{Aff}^1$$ ), giving rise to a notion of twisted monogenerators. In particular, we show a number ring A has class number one if and only if each twisted monogenerator is in fact a global monogenerator $$\theta $$ .

Singular rational curves on elliptic K3 surfaces

AbstractWe show that on every elliptic K3 surface there are rational curves such that , that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to is dense in the Zariski topology. As an application, we give a simple proof of a theorem of Kobayashi in the elliptic case, that is, there are no globally defined symmetric differential forms.

Continuous K-theory and cohomology of rigid spaces

We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the dimension. Likewise, the cohomology groups vanish in degrees above the dimension. The main result provides the existence of an isomorphism between the lowest possibly non-vanishing continuous K-group and the highest possibly non-vanishing cohomology group with integral coefficients. A key role in the proof is played by a comparison between cohomology groups of an admissible Zariski-Riemann space with respect to different topologies; namely, the rh-topology which is related to K-theory as well as the Zariski topology whereon the cohomology groups in question rely.

Spectrum of Zariski Topology in Multiplication Krasner Hypermodules

In this paper, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings. In particular, we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hypermodules. Moreover, we provide some relevant properties of the hypermodule in this topological hyperspace.