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The Diamond Lemma for Power Series Algebras

The main result in this thesis is the generalisation of Bergman's diamond lemma for ring theory to power series rings. This generalisation makes it possible to treat problems in which there arise infinite descending chains. Several results in the literature are shown to be special cases of this diamond lemma and examples are given of interesting problems which could not previously be treated. One of these examples provides a general construction of a normed skew field in which a custom commutation relation holds. There is also a general result on the structure of totally ordered semigroups, demonstrating that all semigroups with an archimedean element has a (up to a scaling factor) unique order-preserving homomorphism to the real numbers. This helps analyse the concept of filtered structure. It is shown that whereas filtered structures can be used to induce pretty much any zero-dimensional linear topology, a real-valued norm suffices for the definition of those topologies that have a reasonable relation to the multiplication operation. The thesis also contains elementary results on degree (as of polynomials) functions, norms on algebras (in particular ultranorms), (Birkhoff) orthogonality in modules, and construction of semigroup partial orders from ditto quasiorders.

Identiferoai:union.ndltd.org:UPSALLA1/oai:DiVA.org:umu-92
Date January 2002
CreatorsHellström, Lars
PublisherUmeå universitet, Matematiska institutionen, Umeå : Umeå universitet
Source SetsDiVA Archive at Upsalla University
LanguageEnglish
Detected LanguageEnglish
TypeDoctoral thesis, monograph, info:eu-repo/semantics/doctoralThesis, text
Formatapplication/pdf
Rightsinfo:eu-repo/semantics/openAccess
RelationDoctoral thesis / Umeå University, Department of Mathematics, 1102-8300 ; 23

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