Topic
Noetherian topological space
About: Noetherian topological space is a research topic. Over the lifetime, 8 publications have been published within this topic receiving 117 citations. The topic is also known as: Noetherian space.
Papers
TL;DR: In this paper, an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple leftR-modules (or, more generally, simple objects in a complete abelian category), is studied.
Abstract: LetR 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 leftR-modules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and whenR is left noetherian the corresponding topological space is noetherian. IfR is commutative (or PI, or FBN) the corresponding topological space is naturally homeomorphic to the maximal spectrum, equipped with the Zariski topology. WhenR is the first Weyl algebra (in characteristic zero) we obtain a one-dimensional irreducible noetherian topological space. Comparisons with topologies induced from those on A. L. Rosenberg’s spectra are briefly noted.
4 citations
25 Sep 2007
TL;DR: The main purpose of as mentioned in this paper is to specify the topological dimensions of X, where X is a Noetherian topological space, and compare them with those of (M).
Abstract: Let R be a commutative ring and let M be an R-module. Let X = Spec(M) be the prime spectrum of M with Zariski topology. Our main purpose in this paper is to specify the topological dimensions of X, where X is a Noetherian topological space, and compare them with those of topological dimensions of (M). Also we will give a characterization for the irreducibility of X and we obtain some related results.
4 citations
Posted Content•
TL;DR: In this article, the authors define a class of sites for which a generalized version of the Brown-Gersten approach works for simplicial pre-sheaves on a Noetherian topological space.
Abstract: There are two approaches to the homotopy theory of simplicial (pre-)sheaves. One developed by Joyal and Jardine works for all sites but produces a model structure which is not finitely generated even in the case of sheaves on a Noetherian topological space. The other one developed by Brown and Gersten gives a nice model structure for sheaves on a Noetherian space of finite dimension but does not extend to all sites. In this paper we define a class of sites for which a generalized version of the Brown-Gersten approach works.
2 citations
TL;DR: In this paper, the authors describe the connectedness dimension of X in terms of Krull dimension of some quotient of M and prove that c(Spec R (M)) = c(Supp(M)).
Abstract: Let R be a commutative Noetherian ring and let M be a finitely generated R-module. Let X = SpecR(M) be the topological space with Zariski topology. Our main goal in this paper is to describe the connectedness dimension of X in terms of Krull dimension of some quotient of M and prove that c(Spec R (M)) = c(Supp(M)). Throughout this paper R denotes a commutative ring with an identity and the notation "‰" denotes the strict inclusion. Let M be an R-module. Then a proper submodule N of M is said to be prime if for any r 2 R and any m 2 M with rm 2 N we have m 2 N or r 2 (N :R M). Further, the spectrum of M is denoted by Spec R (M) and defined by
1 citations
Posted Content•
TL;DR: In this article, an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left R-modules (or, more generally, simple objects in a complete abelian category), is studied.
Abstract: 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 R-modules (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 (or PI, or FBN) the topology is equivalent to the Zariski topology, and when R is the first Weyl algebra (in characteristic zero) we obtain a one-dimensional irreducible noetherian topological space. Comparisons with topologies induced from those on A. L. Rosenberg's spectra are briefly noted.
1 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2013 | 1 |
| 2010 | 1 |
| 2008 | 1 |
| 2007 | 1 |
| 2006 | 1 |