Grupa obrnljivih elementov kolobarja [Z] n [spodaj]

magistrsko delo

Tilen Miklavec (Author), Primož Šparl (Mentor)

Abstract

V magistrskem delu proučujemo kolobarje in njihove grupe obrnljivih elementov. Obrnljivi elementi poljubnega kolobarja z enico namreč tvorijo grupo za pripadajočo multiplikativno operacijo. V posebnem primeru kolobarja Z_n gre za grupo obrnljivih elementov U_n, ki ji rečemo tudi grupa enot. Glavni namen magistrskega dela je predstaviti kolobar Z_n in ugotoviti, kateri znani grupi je izomorfna grupa obrnljivih elementov U_n. Zanima nas tudi, kdaj je ta grupa ciklična. V ta namen natančno opišemo strukturo grupe obrnljivih elementov U_n in jo zapišemo kot direktni produkt samih cikličnih grup. V magistrskem delu pokažemo tudi, kako si lahko z rezultati o strukturi grupe U_n pomagamo pri reševanju nekaterih kongruenčnih enačb.

Keywords

kolobarji;grupa obrnljivih elementov;ciklična grupa;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[T. Miklavec]
UDC:	51(043.2)
COBISS:	12561225
Views:	524
Downloads:	86
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The group of units of the ring Z_n
Secondary abstract:	In the master’s thesis we study finite rings and their groups of units. The invertible elements of an arbitrary ring with a multiplicative identity form a group for the corresponding multiplicative operation. In the special case of the ring Z_n, this is a group denoted by U_n, also called the group of units. The main purpose of the thesis is to introduce the ring Z_n and to determine, for each integer n, the well known group that the group of units U_n is isomorphic to. We also determine the necessary and suficient condition on n for the group U_n to be cyclic. To this end, we investigate the structure of the group of units U_n and show that we can write it as a direct product of certain cyclic groups. We also indicate how our results can be used for solving some congruence equations.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje
Pages:	II, 53 str.
ID:	11206310