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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Lehtikangas, Vilppu", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-03-08T08:32:18Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-03-08T08:32:18Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/74515", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on rakentaa moduulien teoria ryhm\u00e4- ja rengasteorian alkeista l\u00e4htien, sek\u00e4 osoittaa p\u00e4\u00e4ideaalialueiden moduulien p\u00e4\u00e4lause.\nModuuli on joukko G varustettuna yhteenlaskutoimituksella, joka tekee siit\u00e4 abelin ryhm\u00e4n, sek\u00e4 toiminnaksi kutsutulla kuvauksella joka liitt\u00e4\u00e4 jokaiseen G:n kertoimien renkaan alkioon ja G:n alkioon jonkin G:n alkion. Toiminnan m\u00e4\u00e4ritell\u00e4\u00e4n my\u00f6s toteuttavan vektoriavaruuksien tunnetut distributiivisuusominaisuudet, jolloin se yleist\u00e4\u00e4 vektoriavaruuden skalaaritulon k\u00e4sitteen yleiselle renkaalle. N\u00e4in ollen vektoriavaruuden yleisen m\u00e4\u00e4ritelm\u00e4n nojalla vektoriavaruudet ovat t\u00e4sm\u00e4lleen kuntakertoimisia moduuleja. Osoittautuu, ett\u00e4 my\u00f6s abelin ryhm\u00e4t ja renkaat ovat moduuleja, joiden kerroinrenkaina ovat vastaavasti kokonaislukujen rengas, sek\u00e4 rengas itse. Tulemme huomaamaan, ett\u00e4 monet ryhm\u00e4- ja rengasteorian tuloksista yleistyv\u00e4t my\u00f6s moduuleille.\nTutkimme, kuinka moduulin kerroinrenkaan rakenne vaikuttaa itse moduulin ominaisuuksiin. Tulemme osoittamaan, ett\u00e4 p\u00e4\u00e4ideaalialueitten tapauksessa jokainen p\u00e4\u00e4ideaalialueen moduuli voidaan esitt\u00e4\u00e4 yksik\u00e4sitteisesti suorana summana vapaasta moduulista, eli moduulista jolla on vektoriavaruuden tavoin kanta, sek\u00e4 \u00e4\u00e4rellisen monesta syklisest\u00e4 tekij\u00e4moduulista. Tarkastelemme lopuksi lyhyesti joitain t\u00e4m\u00e4n p\u00e4\u00e4ideaalimoduulien p\u00e4\u00e4lauseeksi kutsutun tuloksen sovelluksia, kuten \u00e4\u00e4rellisesti viritettyjen abelin ryhmien p\u00e4\u00e4lausetta, sek\u00e4 neli\u00f6matriisin Jordanin kanonista muotoa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-03-08T08:32:18Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-03-08T08:32:18Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "75", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "moduuli", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ryhm\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "rengas", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ideaali", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "p\u00e4\u00e4ideaalialue", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "P\u00e4\u00e4ideaalialueen moduulien p\u00e4\u00e4lause", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202103081872", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "abstrakti algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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