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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pirnes, Erika", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-06-15T11:35:57Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-06-15T11:35:57Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/58598", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Olkoon R kommutatiivinen rengas. T\u00e4m\u00e4n tutkielman tarkoituksena on etsi\u00e4 yl\u00e4- ja\r\nalarajat \u00e4\u00e4rellisviritteisen ideaalin I = (a1, . . . , an) \u2282 R minimaaliselle viritt\u00e4j\u00e4m\u00e4\u00e4r\u00e4lle.\r\nT\u00e4rke\u00e4n\u00e4 ty\u00f6kaluna toimii moduliteoria; modulit yleist\u00e4v\u00e4t sek\u00e4 ideaalit ett\u00e4\r\nvektoriavaruudet.\r\nJos joukko {a1, . . . , an} on vektoriavaruuden V viritt\u00e4j\u00e4joukko, jossa mik\u00e4\u00e4n alkioista\r\nai ei kuulu toisten viritt\u00e4jien lineaariseen verhoon, on kyseinen joukko lineaarisesti\r\nriippumaton viritt\u00e4j\u00e4joukko eli kanta. T\u00e4ll\u00f6in kaikissa vektoriavaruuden V viritt\u00e4j\u00e4joukoissa\r\non v\u00e4hint\u00e4\u00e4n n alkiota, ja kaikissa kannoissa niit\u00e4 on tasan n kappaletta.\r\nTarpeettomien viritt\u00e4jien poistaminen ei ideaalin ollessa kyseess\u00e4 kuitenkaan riit\u00e4.\r\nVaikka mit\u00e4\u00e4n ideaalin viritt\u00e4jist\u00e4 ai ei voitaisi poistaa, pienempi viritt\u00e4j\u00e4joukko\r\nsaattaa silti olla olemassa.\r\nErityinen kokoelma renkaita, joissa ideaalin minimaalisen viritt\u00e4j\u00e4m\u00e4\u00e4r\u00e4n selvitt\u00e4minen\r\non verrattain helppoa, on lokaalit renkaat. Hieman yleisemmin: kun R on lokaali\r\nrengas, niin \u00e4\u00e4rellisviritteisen R-modulin minimaalinen viritt\u00e4j\u00e4m\u00e4\u00e4r\u00e4 on sama kuin\r\ntietyn renkaaseen ja moduliin liittyv\u00e4n vektoriavaruuden dimensio. Todistus pohjautuu\r\nmoduliteorian tulokseen, joka tunnetaan nimell\u00e4 Nakayaman lemma. Lokaalien\r\nrenkaiden tapauksessa kysymys voidaan siten palauttaa vektoriavaruuden dimension\r\nselvitt\u00e4miseen.\r\nRenkaan lokalisaatio syntyy samantapaisella (vaikkakin hieman yleisemm\u00e4ll\u00e4) konstruktiolla\r\nkuin rationaaliluvut. Sen avulla voidaan l\u00f6yt\u00e4\u00e4 alaraja ideaalin minimaaliselle\r\nviritt\u00e4j\u00e4m\u00e4\u00e4r\u00e4 alle renkaassa, joka ei ole lokaali. Jokaisella ideaalilla on lokalisaatiossa\r\nsit\u00e4 vastaava ideaali, jota kutsutaan sen laajennukseksi, ja laajennuksen\r\nminimaalinen viritt\u00e4j\u00e4m\u00e4\u00e4r\u00e4 on pienempi tai yht\u00e4 suuri kuin alkuper\u00e4isen ideaalin\r\nminimaalinen viritt\u00e4j\u00e4m\u00e4\u00e4r\u00e4. Alkuideaalin suhteen tehty lokalisaatio on lokaali rengas,\r\njoten yll\u00e4 esitetty tulos antaa halutun alarajan. Jos t\u00e4m\u00e4n suuruinen viritt\u00e4j\u00e4joukko\r\non l\u00f6ydetty, voidaan n\u00e4in todistaa ett\u00e4 se on minimaalinen siin\u00e4 mieless\u00e4, ett\u00e4\r\npienempi\u00e4 viritt\u00e4j\u00e4joukkoja ei ole olemassa.\r\nMik\u00e4li R on Noetherin rengas, voidaan sen ideaalien minimaaliselle viritt\u00e4j\u00e4m\u00e4\u00e4r\u00e4lle\r\nl\u00f6yt\u00e4\u00e4 my\u00f6s yl\u00e4raja. T\u00e4ss\u00e4 tekstiss\u00e4 esitell\u00e4\u00e4n Otto Forsterin tulos. Jokaiselle \u00e4\u00e4rellisviritteiselle\r\nR-modulille E m\u00e4\u00e4ritell\u00e4\u00e4n luku b(E) siten, ett\u00e4 E voidaan viritt\u00e4\u00e4\r\njoukolla alkioita, joita on b(E) kappaletta. My\u00f6s t\u00e4m\u00e4 tulos hy\u00f6dynt\u00e4\u00e4 lokalisaatiota,\r\nja sen lis\u00e4ksi Krullin dimensiok\u00e4sitett\u00e4 ja Zariski-topologiaa.\r\nK\u00e4sitteiden selvent\u00e4miseksi k\u00e4ytetyist\u00e4 esimerkeist\u00e4 suurin osa k\u00e4sittelee polynomirenkaita.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2018-06-15T11:35:56Z\r\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-06-15T11:35:57Z (GMT). No. of bitstreams: 0\r\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "64", "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": "eng", "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": "kommutatiivinen 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": "lokalisaatio", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "viritt\u00e4j\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "moduli", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "The minimal number of generators for ideals in commutative rings", "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-201806153244", "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": "polynomit", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "polynomials", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": "", "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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