Let $R$ be a $G$-graded ring and M $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to set all submodules $Q$ such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we topology on $\mathcal{PS}_G(M)$ having Zariski prime $Spec_G(M)$ as subspace topology, investigate several topological properties space.

There is a close relationship between the embedded topology of complex plane curves and (group-theoretic) arithmetic elliptic curves. In recent paper, we studied some arrangements that include special smooth component, via torsion properties induced by divisors in curve associated to remaining components, which an property. When this has maximal flexes, there natural isomorphism its Jacobian va...

1 Affine schemes 4 1.1 Motivation and review of varieties . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 First attempt at defining an affine scheme . . . . . . . . . . . . . . . . . . . 4 1.3 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 The ringed space s...

A meadow is a zero totalised field (0 = 0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational axiomatization of these operators and thus obtain a finite basis for differential cancellation meadows. Using the Zariski topology we prove the existence of a d...

We show that if a group automorphism of Cremona arbitrary rank is also homeomorphism with respect to either the Zariski or Euclidean topology, then it inner up field base-field. Moreover, we similar result holds consider groups polynomial automorphisms affine spaces instead groups.

The nature of finitely generated infinite index subgroups of SL(3,Z) remains extremely mysterious. It follows from the famous theorem of Tits [12] that free groups abound and, moreover, Zariski dense free groups abound. Less trivially, classical arithmetic considerations (see for example §6.1 of [9]) can be used to construct surface subgroups of SL(3,Z) of every genus ≥ 2. However these are con...

Abstract The aim of this paper is to investigate the behaviour prime and semiprime subgroups groups, their relation with existence abelian normal subgroups. In particular, we study set Spec( G ) all a group endowed Zariski topology and, among other examples, construct an infinite whose proper are form descending chain type ? + 1.

Let $$C \rightarrow \mathop {{\mathrm{Spec}}}\nolimits (R)$$ be a relative proper flat curve over henselian base. G reductive C-group scheme. Under mild technical assumptions, we show that G-torsor C which is trivial on the closed fiber of locally for Zariski topology.