Ideal spaces: An extension of structure spaces of rings

TL;DR

This paper develops a generalized theory of ideal spaces for arbitrary classes of ring ideals, extending classical structure spaces and exploring topological properties related to ring idempotents and connectivity.

Contribution

It introduces a unified framework for ideal spaces encompassing various ideal types and characterizes their topological properties, including sobriety and disconnectedness.

Findings

Characterization of sober ideal spaces.

Introduction of strongly disconnected spaces and their implications.

Conditions for spectrum connectivity.

Abstract

Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly irreducible, irreducible, completely irreducible, proper, minimal, primary, nil, nilpotent, regular, radical, principal, finitely generated ideals. We characterise ideal spaces that are sober. We introduce the notion of a strongly disconnected spaces and show that for a ring with zero Jacobson radical, strongly disconnected ideal spaces containing all maximal ideals of the ring imply existence of non-trivial idempotent elements in the ring. We also give a sufficient condition for a spectrum to be connected.