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Thesis: Maximal ideals of rings in models of set theory

Glas, P. (2017) Thesis: Maximal ideals of rings in models of set theory. Bachelor's Thesis, Mathematics.

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Abstract

In this thesis we investigate a method used in set theory, namely Paul Cohen's forcing technique. The forcing technique allows one to obtain models of Zermelo-Fraenkel set theory such that the Axiom of Choice (AC) fails. These models are called symmetric extensions. Properties of algebraic structures that crucially depend on AC do not hold in a symmetric extension. An example of this follows from a theorem proved by Wilfred Hodges in 1973. This theorem states that if every commutative ring with 1 has a maximal ideal, than AC holds. Thus any symmetric extension contains a commutative ring with 1 that has no maximal ideal. A natural question is whether it is possible to find explicitly a commutative ring with 1 that has no maximal ideal in a particular symmetric extension. In this thesis we show that this is the case for a particular symmetric extension known as the Basic Cohen model. First, we study Hodges' proof in detail. After that we introduce the forcing method and study a particular symmetric extension. In this extension we describe a commutative ring with 1 that has no maximal ideal, by using a ring which is essential in Hodges' proof.

Item Type: Thesis (Bachelor's Thesis)
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 08:26
Last Modified: 15 Feb 2018 08:26
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/14870

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