Vsote izjemnih enot

delo diplomskega seminarja

Ajda Lemut (Author), David Dolžan (Mentor)

Abstract

Element $u$ iz kolobarja je izjemna enota, če sta $u$ in $1-u$ enoti, torej če sta $u$ in $1-u$ obrnljiva. V delu se najprej posvetimo kolobarjem ostankov ${\mathbb Z}_n$, nato pa sledi posplošitev na poljubne končne komutativne kolobarje z enico. V obeh primerih najprej dokažemo formulo za izračun števila izjemnih enot, nato pa še formulo za izračun predstavitev poljubnega elementa iz kolobarja kot vsoto $k$ izjemnih enot.

Keywords

matematika;izjemne enote;kolobar ostankov;končni kolobarji;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[A. Lemut]
UDC:	512
COBISS:	76460291
Views:	591
Downloads:	52
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Sums of exceptional units
Secondary abstract:	Element $u$ from some ring is an exceptional unit if both $u$ and $1-u$ are units, so if both $u$ and $1-u$ are invertible. In this work we first focus on the residue class rings modulo $n$, and then generalize it to all finite commutative rings with identity. In both cases, we first prove the formula for calculating the number of exceptional units, and then the formula for calculating the representations of any element in the ring as the sum of $k$ exceptional units.
Secondary keywords:	mathematics;exceptional units;residue class ring;finite rings;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	31 str.
ID:	13411328