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 ZermeloFraenkel 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:  https://fse.studenttheses.ub.rug.nl/id/eprint/14870 
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