Dokument: Coprime Modules and Comodules
| Titel: | Coprime Modules and Comodules | |||||||
| URL für Lesezeichen: | https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=3431 | |||||||
| URN (NBN): | urn:nbn:de:hbz:061-20060627-001431-9 | |||||||
| Kollektion: | Dissertationen | |||||||
| Sprache: | Englisch | |||||||
| Dokumententyp: | Wissenschaftliche Abschlussarbeiten » Dissertation | |||||||
| Medientyp: | Text | |||||||
| Autor: | Wijayanti, Indah Emilia [Autor] | |||||||
| Dateien: |
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| Beitragende: | Prof. Dr. Wisbauer, Robert [Gutachter] Prof. Dr. Kerner, Otto [Gutachter] | |||||||
| Stichwörter: | Primmoduln, Koprimmoduln, Primkomoduln, Koprimkomoduln, KoprimkoalgebrenPrimeModules, CoprimeModules, PrimeComodules, CoprimeComodules, CoprimeCoalgebras | |||||||
| Dewey Dezimal-Klassifikation: | 500 Naturwissenschaften und Mathematik » 510 Mathematik | |||||||
| Beschreibung: | The study of coalgebras
was motivated by the existing theory of algebras and rings, and by transferring the corresponding knowledge from algebras to coalgebras and from modules to comodules. However, the notion of primeness for rings and modules did not find an adequate counterpart in the coalgebraic setting due to the finiteness theorem for comodules. Only few papers were dealing with this problem. Recall the classical definitions of primeness. If for any non-zero fully invariant submodule $K$ of an $R$-module $ M$, $\Ann_{R}(K) = \Ann_{R}(M)$, then $M$ is called prime; if $M$ is $K$-cogenerated, then $M$ is called fully prime; if $M \in \sigma[K]$, then $M$ is called strongly prime. Dualizing primeness condition, coprimeness can be defined for modules and algebras. If for any proper fully invariant submodule $K$ of an $R$-module $M$, $\Ann_{R}(M/K) = \Ann_{R}(M)$, then $M$ is called coprime; if $M$ is $M/K$-generated, then $M$ is called fully coprime; if $M \in \sigma[M/K]$, then $M$ is called strongly coprime. We investigate the resulting notions for modules and then transfer the outcome to comodules and coalgebras over commutative rings. Notice that for any algebra $A$, coprimeness as an $A$-module implies that $A$ is a simple algebra, but for a coalgebra $C$ the condition to be coprime as a comodule is not so restrictive. Also of interest is the primeness of the endomorphism ring of an $R$-module $M$ and its relationship with the primeness or coprimeness of $M$. In the case of a coalgebra $C$, the comodule endomorphism ring $\End^{C}(C)$ of $C$ is isomorphic to the dual algebra $C* =\Hom_{R}(C,R)$ with the convolution product. In this situation the question reduces to the interplay of primeness and coprimeness conditions of the coalgebra $C$ and the dual algebra $C*$. If $M$ is prime and has a non-zero socle, then $\overline{R} := R/\Ann_{R}(M)$ is a left primitive ring. Similarly, if $M$ is coprime and $\Rad(M) \neq M$, then $\overline{R} := R/\Ann_{R}(M)$ is a primitive ring. We apply these results to $C$-comodules, where $C$ is a coalgebra over a commutative ring, and consider these conditions for the coalgebra $C$ itself. If $C$ is prime as a right $C$-comodule and $\Soc(C) \neq 0$, then $\Cst$ is a simple algebra and finitely generated as an $R$-module. If $C$ is coprime as a right $C$-comodule and $\Rad(C) \neq C$, then we have the same structure of $C*$. Studies of localization and colocalization of coalgebras over a field were done by some authors with respect to coidempotent subcoalgebras of $C$. To avoid the dependence on the base ring being a field, we give an outline of colocalization in module categories and then apply it to comodules and coalgebras. In abelian categories the existence of a colocalization functor depends on the presence of enough projectives in the category. We transfer the technique of colocalization in the category of $R$-modules to the comodule situation. For cohereditary torsion theories induced by some $C$-comodule $P$ which is finitely generated and projective as $\Cst$-module, a coalgebra structure can be defined on $P \ots P^{*}$ and the colocalization $P \ots P^{*} \lra C$ is a coalgebra morphism. In module categories of type $\sM$ the torsion theory induced by the injective hull of $M$ is of particular interest and primeness of the module leads to a special structure of the module of quotient. Thus the question arises about the role of a projective hull of a subgenerator in the dual case. However no comparable constructions are possible in this situation. In fact, the existence of a projective hull $P \lra C$ of a strongly coprime coalgebra implies that $\Rad(P) = 0$ and $C \simeq P$. | |||||||
| Lizenz: |  Urheberrechtsschutz | |||||||
| Fachbereich / Einrichtung: | Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik | |||||||
| Dokument erstellt am: | 27.06.2006 | |||||||
| Dateien geändert am: | 12.02.2007 | |||||||
| Promotionsantrag am: | 27.06.2006 | |||||||
| Datum der Promotion: | 27.06.2006 |
