$k$-spaces of non-domain-valued geometric points

TL;DR

This paper explores the topological structure of algebraic sets with zero divisors through a new $k$-space topology, generalizing Zariski spaces and revealing properties like compactness and connectedness.

Contribution

It introduces a novel $k$-space topology on proper ideals of $k$-algebras, extending the classical Zariski space framework.

Findings

$k$-spaces are $T_0$, quasi-compact, spectral, and connected

Continuous maps between $k$-spaces are studied

Raises questions about sheaf constructions analogous to affine schemes

Abstract

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a $k$-algebra and this new ``$k$-space'' becomes a generalization of the corresponding Zariski space. We prove that a $k$-space is $T_{\scriptscriptstyle 0}$, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such $k$-spaces. We conclude with a question about construction of a sheaf of $k$-spaces similar to affine schemes.