How to Expand the Zariski Topology
نویسنده
چکیده
We introduce the notion of a Hu-Liu prime ideal in the context of left commutative rngs, and establish the contravariant functor from the category of left commutative rngs into the category of topological spaces. It is well known that new points must be introduced in order to expand algebraic geometry over algebraically closed fields into Grothendieck’s scheme theory over commutative rings. We believe that the idea of adding new points to an old space is still essential for the attempt to expand algebraic geometry over algebraically closed fields into a kind of geometry over a class of non-commutative rings. Clearly, whether we can use the natural idea successfully depends on whether we can find new points satisfactorily. Since points in Grothendieck’s scheme theory are prime ideals, the problem of finding satisfactory new points is how to choose some classes of rings to get a satisfactory generalization of prime ideals. The purpose of this paper is to give a solution to the problem. Our solution is based on the notion of the additive halo introduced in [4], which comes from some facts obtained in our attempt to generalize the Lie correspondence between connected linear Lie groups and linear Lie algebras. After choosing a class of rings called left commutative rngs, we introduce the notion of a Hu-Liu prime ideal, characterize the nil radical by using Hu-Liu prime ideals, and establish the contravariant functor from the category of left commutative rngs into the category of topological spaces.
منابع مشابه
The Graded Classical Prime Spectrum with the Zariski Topology as a Notherian Topological Space
Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...
متن کاملOn two problems concerning the Zariski topology of modules
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...
متن کاملPRIMARY ZARISKI TOPOLOGY ON THE PRIMARY SPECTRUM OF A MODULE
Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. We define the primary spectrum of $M$, denoted by $mathcal{PS}(M)$, to be the set of all primary submodules $Q$ of $M$ such that $(operatorname{rad}Q:M)=sqrt{(Q:M)}$. In this paper, we topologize $mathcal{PS}(M)$ with a topology having the Zariski topology on the prime spectrum $operatorname{Spec}(M)$ as a sub...
متن کاملInverse topology in BL-algebras
In this paper, we introduce Inverse topology in a BL-algebra A and prove the set of all minimal prime filters of A, namely Min(A) with the Inverse topology is a compact space, Hausdorff, T0 and T1-Space. Then, we show that Zariski topology on Min(A) is finer than the Inverse topology on Min(A). Then, we investigate what conditions may result in the equivalence of these two topologies. Finally,...
متن کاملSPECTRUM OF PRIME FUZZY HYPERIDEALS
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.
متن کامل