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master's thesis
Komutativni prsteni
Zagreb: University of Zagreb, Faculty of Science, 2014. urn:nbn:hr:217:318988
Zrno, Marina
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Zrno, M. (2014). Komutativni prsteni (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:318988

Zrno, Marina. "Komutativni prsteni." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:318988

Zrno, Marina. "Komutativni prsteni." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:318988

Zrno, M. (2014). 'Komutativni prsteni', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:318988

Zrno M. Komutativni prsteni [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2014 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:318988

M. Zrno, "Komutativni prsteni", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2014. Available at: https://urn.nsk.hr/urn:nbn:hr:217:318988

Metadata
TitleKomutativni prsteni
AuthorMarina Zrno
MentorOzren Perše (mentor)
Committee memberOzren Perše (predsjednik povjerenstva)
Committee memberIvica Nakić (član povjerenstva)
Committee memberZvonimir Tutek (član povjerenstva)
Committee memberVedran Krčadinac (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2014-07-14, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Centralni pojam ovog diplomskog rada je komutativan prsten, jedna od osnovnih algebarskih struktura u matematici. U Poglavlju 1 uvodimo neke osnovne definicije koje će nam biti potrebne za daljnje analize, kao na primjer, domene i polje razlomaka. Poseban naglasak smo stavili na prsten polinoma. Poglavlje 2 počinje s definicijom polinoma, vodećih koeficijenata polinoma, stupnja polinoma i prstena polinoma. U Poglavlju 3 ćemo vidjeti da, kada je k polje, svi poznati teoremi koji vrijede u Z; imaju analogon u k[x]; štoviše, vidjet ćemo da se svi poznati dokazi mogu prenijeti ovdje. Prezentiramo algoritam dijeljenja za polinome s koeficijentima iz polja. U Poglavlju 4 ćemo promatrati ona preslikavanja koja 'čuvaju strukturu'; tj., preslikavanja među prstenima koja respektiraju obje operacije, zbrajanje i množenje u prstenu. Zatim uvodimo pojam ideala u prstenu, koji čini centralno mjesto u teoriji prstena. U Poglavlju 5 uvodimo pojam Euklidovog prstena i donosimo važan teorem koji kaže da je svaki Euklidov prsten domena glavnih ideala. U Poglavlju 6 definiramo kvocijentni prsten i navodimo Prvi teorem o izomorfizmu za prstene. U Poglavlju 7 i Poglavlju 8 uvodimo dva zanimljiva tipa ideala: proste ideale i maksimalne ideale. Onda ćemo, nakon što uvedemo pojam domene s jedinstvenom faktorizacijom, dokazati da je svaka domena glavnih ideala domena s jedinstvenom faktorizacijom.
Abstract (english)
The main focus of this work are commutative rings, which are one of the basic algebraic structures in mathematics. In Chapter 1 we introduce some basic definitions that will be needed for further analysis, like for example, domains and fraction fields. In this work we give a special attention to the ring of polynomials. Chapter 2 begins with definitions of polynomial, leading coefficient of polynomial, degree of polynomial and ring of polynomials. In Chapter 3 we are going to see that, when k is a field, virtually all the familiar theorems valid in Z have polynomial analogs in k[x]; moreover, we shall see that the familiar proofs can be translated into proofs here. We present division algorithm for polynomials with coefficients in a field. In Chapter 4 we observe those mappings that preserve the structure, the (ring) homomorphism. Afterwards we introduce the notion of ideals that makes the central point in the theory of rings. Chapter 5 defines euclidean ring and proves that every euclidean ring is a principal ideal domain. Chapter 6 defines quotient ring and proves first isomorphism theorem. Chapter 7 and Chapter 8 introduce two interesting type of ideals: prime and maximal ideals in commutative rings, and define unique factorization domains. Afterwards we are going to prove a common generalization: Every principal ideal domain has a unique factorization.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:318988
Study programmeTitle: Mathematical Statistics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-02-20 14:26:50