On the theory of Krull rings and injective modules

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dc.contributor.advisorHughes, Kenneth Ren_ZA
dc.contributor.authorPrince, R Nen_ZA
dc.date.accessioned2016-10-19T03:52:27Z
dc.date.available2016-10-19T03:52:27Z
dc.date.issued1988en_ZA
dc.description.abstractIn 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.en_ZA
dc.identifier.apacitationPrince, R. N. (1988). <i>On the theory of Krull rings and injective modules</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/22179en_ZA
dc.identifier.chicagocitationPrince, R N. <i>"On the theory of Krull rings and injective modules."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1988. http://hdl.handle.net/11427/22179en_ZA
dc.identifier.citationPrince, R. 1988. On the theory of Krull rings and injective modules. University of Cape Town.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Prince, R N AB - 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. DA - 1988 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1988 T1 - On the theory of Krull rings and injective modules TI - On the theory of Krull rings and injective modules UR - http://hdl.handle.net/11427/22179 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/22179
dc.identifier.vancouvercitationPrince RN. On the theory of Krull rings and injective modules. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1988 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/22179en_ZA
dc.language.isoengen_ZA
dc.publisher.departmentDepartment of Mathematics and Applied Mathematicsen_ZA
dc.publisher.facultyFaculty of Scienceen_ZA
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherMathematicsen_ZA
dc.subject.otherKrull ringsen_ZA
dc.subject.otherInjective modules (Algebra)en_ZA
dc.titleOn the theory of Krull rings and injective modulesen_ZA
dc.typeMaster Thesis
dc.type.qualificationlevelMasters
dc.type.qualificationnameMScen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
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