<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

An overview of the braid group techniques in the theory of algebraic surfaces from Zariski to the latest results is presented. An outline of the Van Kampen algorithm for computing fundamental groups of complements of curves and the modification of Moishezon-Teicher regarding branch curves of generic projections are given. The paper also contains a description of a quotient of the braid group, n...

Let R be a commutative hyperring with identity. We introduceand study prime fuzzy hyperideals of R. We investigate the Zariski topologyon FHspec(R), the spectrum of prime fuzzy hyperideals of R.

Let G be a reductive algebraic group over an algebraically closed field k. An algebraic characteristic class of degree i for principal G-bundles on schemes is a function c assigning to each principal G-bundle E → X an element c(E) in the Chow group AX, natural with respect to pullbacks. These classes are analogous to topological characteristic classes (which take values in cohomology), and two ...

3 Topics in Commutative Algebra 2 3.1 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Quotient Rings and Homomorphisms . . . . . . . . . . . . . . 5 3.4 The Characteristic of a Ring . . . . . . . . . . . . . . . . . . . 7 3.5 Polynomial Rings in Several Variables . . . . . . . . . . . . . . 7 3.6...

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of ...

We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with (N,+,×) if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in Z. Notably, by constructing a nontrivial...

In general, the sheaf criterion on the étale topology may be difficult to verify directly, as a scheme will in general have many étale covers. It is clear that a necessary condition for a presheaf F to be a sheaf on Xet is that it be a sheaf with respect to Zariski covers (i.e., its restriction to Xzar is a sheaf), and that it be a sheaf with respect to one-piece étale covers (V → U) such that ...

A series of Zariski pairs and four Zariski triplets were found by using lattice theory of K3 surfaces. There is a Zariski triplet of which one member is a deformation of another.