The Algebraic Sets in the General Mathematics and Teaching Them

Most of the pictures in general mathematics are algebraic sets. Indeed, even the first figures taught in class 1 of elementary school are already algebraic sets or part of algebraic sets, such as lines and segments. Therefore, knowing with certainty the properties of algebraic sets is very important for good teaching of high school mathematics, and it is essential to teach them better. To give suggestions and help teachers teach mathematics more effectively, in this report, we will present the Zariski topology, some of their most important properties and the methods to teach algebraic sets.

Zariski Topologies on Graded Ideals

In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.

The F- Topology on Space of Zero Dimensional rings

Let R be a subring of a ring T, and let F be a non-principal ultrafilter on the natural numbers IN. We consider properties and applications of a countably compact, Hausdorff topology called the "F-topology" defined on space of all zero-dimensional subring of T that contains a fixed subring R. We show that the F-topology is strictly finer than the Zariski topology. We extend results regarding distinguished spectral topologies on the space of zero-dimensional subring.

Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

We translate some graph properties of 𝔸𝔾(R) and Γ(R) to some topological properties of Zariski topology. We prove that the facts “(1) The zero ideal of R is an anti fixed-place ideal. (2) Min(R) does not have any isolated point. (3) Rad(𝔸𝔾 (R)) = 3. (4) Rad(Γ(R)) = 3. (5) Γ(R) is triangulated (6) 𝔸𝔾 (R) is triangulated.” are equivalent. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dt t (𝔸𝔾 (R)) = |ℬ(R)| and also if in addition |Min(R)| > 2, then dt(𝔸𝔾 (R)) = |ℬ (R)|. Finally, it is shown that dt(𝔸𝔾 (R)) is finite if and only if dt t (𝔸𝔾 (R)) is finite if and only if Min(R) is finite.

SOME REMARKS ON THE CLASSICAL PRIME SPECTRUM OF MODULES

Let R be a commutative ring with identity and let M be an R-module. A proper submodule P of M is called a classical prime submodule if abm ∈ P, for a,b ∈ R, and m ∈ M, implies that am ∈ P or bm ∈ P. The classical prime spectrum of M, Cl.Spec(M), is defined to be the set of all classical prime submodules of M. We say M is classical primefule if M = 0, or the map ψ from Cl.Spec(M) to Spec(R/Ann(M)), defined by ψ(P) = (P : M)/Ann(M) for all P ∈ Cl.Spec(M), is surjective. In this paper, we study classical primeful modules as a generalisation of primeful modules. Also we investigate some properties of a topology that is defined on Cl.Spec(M), named the Zariski topology.

Zariski Geometries and Quantum Mechanics

Model theory is the study of mathematical structures in terms of the logical relationships they define between their constituent objects. The logical relationships defined by these structures can be used to define topologies on the underlying sets. These topological structures will serve as a generalization of the notion of the Zariski topology from classical algebraic geometry. We will adapt properties and theorems from classical algebraic geometry to our topological structure setting. We will isolate a specific class of structures, called Zariski geometries, and demonstrate the main classification theorem of such structures. We will construct some Zariski structures where the classification fails by adding some noncommuting structure to a classical one. Finally we survey an application of these nonclassical Zariski structures to computation of formulas in quantum mechanics using a method of structural approximation developed by Boris Zilber.

On the gamma spectrum of multiplication gamma acts

In this paper, we introduce the concept of topological gamma acts as a generalization of Zariski topology. Some topological properties of this topology are studied. Various algebraic properties of topological gamma acts have been discussed. We clarify the interplay between this topological space's properties and the algebraic properties of the gamma acts under consideration. Also, the relation between this topological space and (multiplication, cyclic) gamma act was discussed. We also study some separation axioms and the compactness of this topological space.

Zariski topology on the spectrum of graded pseudo prime submodules

In this article, we introduce the concept of graded pseudo prime submodules of graded modules that is a generalization of the graded prime ideals over commutative rings. We study the Zariski topology on the graded spectrum of graded pseudo prime submodules. We clarify the relation between the properties of this topological space and the algebraic properties of the graded modules under consideration.