Komutativni prsteni

Zrno, Marina (2014) Komutativni prsteni. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

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.

Item Type: Thesis (Diploma thesis)
Supervisor: Perše, Ozren
Date: 2014
Number of Pages: 37
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 02 Sep 2015 13:01
Last Modified: 02 Sep 2015 13:01
URI: http://digre.pmf.unizg.hr/id/eprint/4213

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