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Kvazieuklidovské obory integrity
Čoupek, Pavel ; Šaroch, Jan (advisor) ; Glivický, Petr (referee)
In this thesis, we present an overview of some of the known facts about k-stage Euclidean and quasi-Euclidean rings and domains, certain generalisations of the concept of Euclidean ring, as well as some new results. Among the new results, the norm-free characterization of k-stage Euclidean rings based on a transfinite construction of k-stage Euclidean ring is fundamental and has many applications. Statements providing a way to construct new k-stage Euclidean rings from other k-stage Euclidean rings recieve special attention (with the integral domain case in mind). Also, we present an example of a 3-stage Euclidean integral domain which we believe is a good candidate for not being 2-stage Euclidean. 1
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Primes in discretely ordered quasi-Euclidean domains
Sgallová, Ester ; Šaroch, Jan (advisor) ; Glivická, Jana (referee)
This thesis studies discretely ordered quasi-Euclidean domains. The goal is to study primes and prime pairs in them and to answer the question, whether there can be a cofinal set of them. The first construction gives a domain that does not have a cofinal set primes. Another construction builds a principal ideal domain, which has a cofinal set of primes, but no two distinct non-standard primes differ by a natural number, so there is not a cofinal set of prime pairs. Furthermore, the thesis describes a construction of a principal ideal domain, whitch has a cofinal set of prime a-pairs for any even positive integer a. 1
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Kvazieuklidovské obory integrity
Čoupek, Pavel ; Šaroch, Jan (advisor) ; Glivický, Petr (referee)
In this thesis, we present an overview of some of the known facts about k-stage Euclidean and quasi-Euclidean rings and domains, certain generalisations of the concept of Euclidean ring, as well as some new results. Among the new results, the norm-free characterization of k-stage Euclidean rings based on a transfinite construction of k-stage Euclidean ring is fundamental and has many applications. Statements providing a way to construct new k-stage Euclidean rings from other k-stage Euclidean rings recieve special attention (with the integral domain case in mind). Also, we present an example of a 3-stage Euclidean integral domain which we believe is a good candidate for not being 2-stage Euclidean. 1
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