Prsteni polinoma i formalnih redova

master's thesis

• Author: Judita Janković
• Mentor: Goran Muić (mentor)
• Committee member: Goran Muić (predsjednik povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Boris Guljaš (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog diplomskog rada (koji smo podijelili u dva poglavlja) je konstruirati prsten polinoma i formalnih redova u jednoj i više varijabli te proučiti njihova osnovna svojstva. U prvom poglavlju prikazali smo glavne rezultate iz teorije prstena – definicije jednostavnijih algebarskih struktura (grupoida, polugrupa, monoida, grupa) koje smo postupno gradili do osnovne strukture kojom se bavimo u radu – prstena. Osim definicije i primjera prstena, u poglavlju smo još definirali i neke bogatije strukture kao što su tijelo i polje. U drugom poglavlju definirali smo osnovne operacije (zbrajanje i množenje) u prstenu polinoma jedne i više varijabli te u formalnim redovima jedne varijable s ciljem da bismo pokazali da skupovi polinoma i formalnih redova sa definiranim operacijama tvore prsten. Nadalje, proučavali smo međusobne odnose između prstena polinoma i formalnih redova, kao i odnose sa skupovima iz kojih potječu njihovi koeficijenti. Na kraju poglavlja prikazali smo uvjete postojanja i neka svojstva invertibilnih elemenata u prstenu formalnih redova.

Abstract (english)

Goal of this graduate thesis is to construct a ring of polynomials (and formal power series) in one and more indeterminates over a ring \(R\) and to study their main properties. Thesis is divided into two chapters. In the first chapter we show main concepts from ring theory - definitions of algebraic structures (groupoid, semigroup, monoid, group) that we use to construct the ring. Beside the definition and examples of rings, in this chapter we define some structures like division ring and field. In second chapter we define operations (addition and multiplication) in polynomial ring and in formal power series ring (with one and more indeterminates). For those structures and operations, we show that they form a ring. We study relations between ring of polynomials and its superset ring of formal power series, as well as relation with ring from which their coefitients came from. In the end of second chapter we show conditions for existence of a unit and some of the properties of units in formal power series.

• Keywords: polinomi; formalni redovi; teorija prstena
• Keywords (english): polynomials; formal power series; ring theory
• Language: croatian
• URN:NBN: urn:nbn:hr:217:196789
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-05-14 11:46:19