Ideal spaces: An extension of structure spaces of rings
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910570545938432 |
|---|---|
| author | Dube, Themba Goswami, Amartya |
| author_facet | Dube, Themba Goswami, Amartya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_09586 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ideal spaces: An extension of structure spaces of rings Dube, Themba Goswami, Amartya Commutative Algebra Rings and Algebras 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. |
| title | Ideal spaces: An extension of structure spaces of rings |
| topic | Commutative Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2203.09586 |