Affiliation:
1. Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa
2. National Institute for Theoretical and Computational Sciences, South Africa
3. Department of Mathematics and Applied Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa
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 nontrivial idempotent elements in the ring. We also give a sufficient condition for a spectrum to be connected.
Publisher
World Scientific Pub Co Pte Ltd
Subject
Applied Mathematics,Algebra and Number Theory
Cited by
2 articles.
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1. Iséki spaces of semirings;Bollettino dell'Unione Matematica Italiana;2023-04-07
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