Vsote izjemnih enot : delo diplomskega seminarja

Lemut, Ajda (author), Dolžan, David (mentor)

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.

• Language: Slovenian
• Keywords: izjemne enote, kolobar ostankov, končni kolobar
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2021
• PID: 20.500.12556/RUL-130430
• UDC: 512
• COBISS.SI-ID: 76460291
• Publication date in RUL: 15.09.2021

Secondary language

• Language: English
• Title: Sums of exceptional units

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.

• Keywords: exceptional units, residue class ring, finite ring