We prove that the Auslander-Reiten conjecture holds for commutative standard graded artinian algebras, in two situations: The first is under the assumption that the modules considered are graded and generated in a single degree. The second is under the assumption that the algebra is generic Gorenstein of socle degree 3.

The main aim of this article is to study (v, k, λ)-symmetric designs admitting a flag-transitive and point-primitive automorphism group G whose socle is PSL(3, q). We indeed show that the only possible design satisfying these conditions is a Desarguesian projective plane PG(2, q) and G ⩾ PSL(3, q).

Let L be a complete lattice. We introduce and characterize the prime L-submodules of a unitary module over a commutative ring with identity. Finally, we investigate the Zariski topology on the prime L-Spectrum of a unitary module, consisting of the collection of all prime L-submodules, and prove that for L-top modules the Zariski topology on L-Spec(M) exists. © 2007 Elsevier B.V. All rights res...

This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental ...

We continue to study Zariski pairs in sextics. In this paper, we study Zariski pairs of sextics which are not irreducible. The idea of the construction of Zariski partner sextic for reducible cases is quit different from the irreducible case. It is crucial to take the geometry of the components and their mutual intersection data into account. When there is a line component, flex geometry (i.e.,...

Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p > 0, T a maximal torus of G and B a Borel subgroup containing T . Each weight in X(T ) determines a line bundle on the flag varietyG/B. It turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of B-m...

We compute the fundamental groups of all irreducible plane sextics constituting classical Zariski pairs

A new class of noncommutative k-algebras (for k an algebraically closed field) is defined and shown to contain some important examples of quantum groups. To each such algebra, a first order theory is assigned describing models of a suitable corresponding geometric space. Model-theoretic results for these geometric structures are established (uncountable categoricity, quantifier elimination to t...

We describe the structure QHO = QHON (dependent on the positive integer number N) on the universe L which is a finite cover, of order N, of the projective line P = P(F), F an algebraically closed field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension 1. We also prove that QHO is not classical in the sense that the structure is not interpretable in a...