Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left Rmodules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and when R is left noetherian the corresponding topological space is noetherian. If R is commutative (...

There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is natur...

I Koizumi, op. cit., 2, 267. 10 0. Zariski, "The Reduction of the Singularities of an Algebraic Surface," Ann. Math., 40, 639689, 1939; "A Simplified Proof for the Resolution of Singularities of an Algebraic Surface," ibid., 43, 583-593, 1942; S. Abhyankar, "The Theorem of Local Uniformization on Algebraic Surfaces over Modular Fields," Ann. Math. (to appear). "0. Zariski, "Pencils on an Algebr...

Abstract We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.

let $r$ be an associative ring and let $m$ be a left $r$-module.let $spec_{r}(m)$ be the collection of all prime submodules of $m$ (equipped with classical zariski topology). there is a conjecture which says that every irreducible closed subset of $spec_{r}(m)$ has a generic point. in this article we give an affirmative answer to this conjecture and show that if $m$ has a noetherian spectrum, t...