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On the theory of Krull rings and injective modules
In the first chapter we give an outline of classical KRULL rings as in SAMUEL (1964), BOURBAKI (1965) and FOSSUM (1973). In the second chapter we introduce two notions important to our treatment of KRULL theory. The first is injective modules and.the second torsion theories. We then look at injectiv...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2016
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| _version_ | 1867613305248415744 |
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| access_status_str | Open Access |
| author | Prince, R N |
| author2 | Hughes, Kenneth R |
| author_browse | Hughes, Kenneth R Prince, R N |
| author_facet | Hughes, Kenneth R Prince, R N |
| author_sort | Prince, R N |
| collection | Thesis |
| description | In the first chapter we give an outline of classical KRULL rings as in SAMUEL (1964), BOURBAKI (1965) and FOSSUM (1973). In the second chapter we introduce two notions important to our treatment of KRULL theory. The first is injective modules and.the second torsion theories. We then look at injective modules over Noetherian rings as in MATLIS [1958] and then over KRULL rings as in BECK [1971]. We show that for a KRULL ring there is a torsion theory (N,M) where N is the pseudo-zero modules and M the set of N-torsion-free (BECK calls these co-divisorial) modules. From LAMBEK [1971] there is a full abelian sub category C, namely the category of N-torsion-free, N-divisible modules, with exact reflector. We show in C (I) every direct sum of injective modules is injective and (II) C has global dimension at most one. It is these two properties that we exploit in the third chapter to give another characterization of KRULL rings. Then we generalize this to rings with zero-divisors and find that (i) R has to be reduced (ii) the ring is KRULL if and only if it is a finite product of fields and KRULL domains (iii) the injective envelope of the ring is semi-simple artinian. We then generalize the ideas to rings of higher dimension. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/22179 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:34:00.978Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/22179 On the theory of Krull rings and injective modules Prince, R N Hughes, Kenneth R Mathematics Krull rings Injective modules (Algebra) In the first chapter we give an outline of classical KRULL rings as in SAMUEL (1964), BOURBAKI (1965) and FOSSUM (1973). In the second chapter we introduce two notions important to our treatment of KRULL theory. The first is injective modules and.the second torsion theories. We then look at injective modules over Noetherian rings as in MATLIS [1958] and then over KRULL rings as in BECK [1971]. We show that for a KRULL ring there is a torsion theory (N,M) where N is the pseudo-zero modules and M the set of N-torsion-free (BECK calls these co-divisorial) modules. From LAMBEK [1971] there is a full abelian sub category C, namely the category of N-torsion-free, N-divisible modules, with exact reflector. We show in C (I) every direct sum of injective modules is injective and (II) C has global dimension at most one. It is these two properties that we exploit in the third chapter to give another characterization of KRULL rings. Then we generalize this to rings with zero-divisors and find that (i) R has to be reduced (ii) the ring is KRULL if and only if it is a finite product of fields and KRULL domains (iii) the injective envelope of the ring is semi-simple artinian. We then generalize the ideas to rings of higher dimension. 2016-10-19T03:52:27Z 2016-10-19T03:52:27Z 1988 Master Thesis Masters MSc http://hdl.handle.net/11427/22179 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics Krull rings Injective modules (Algebra) Prince, R N On the theory of Krull rings and injective modules |
| thesis_degree_str | Master's |
| title | On the theory of Krull rings and injective modules |
| title_full | On the theory of Krull rings and injective modules |
| title_fullStr | On the theory of Krull rings and injective modules |
| title_full_unstemmed | On the theory of Krull rings and injective modules |
| title_short | On the theory of Krull rings and injective modules |
| title_sort | on the theory of krull rings and injective modules |
| topic | Mathematics Krull rings Injective modules (Algebra) |
| url | http://hdl.handle.net/11427/22179 |
| work_keys_str_mv | AT princern onthetheoryofkrullringsandinjectivemodules |