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Title: Universal fields of fractions : their orderings and determinants
Author: Revesz, Gabor
Awarding Body: University of London
Current Institution: Royal Holloway, University of London
Date of Award: 1981
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We are concerned with two problems. Firstly, given a ring R and an epic R-field K, under what conditions can K be fully ordered? Epic R-fields can be constructed in terms of matrices over R; this makes it natural, in describing full orders on K, to consider matrix cones over R rather then ordinary cones of elements of K. Essentially, a matrix cone over R, associated with a given ordering of K consists of all square matrices which either become singular or have positive Dieudonne determinant over K. We give necessary and sufficient conditions in terms of matrix cones for (i) an epic R-field to be orderable, (ii) a full order on R to be extendible to a field of fractions of R and (iii) for such an extension to be unique. The second problem is finding K1(U(R)), where R is is a Sylvester domain and U(R) denotes its universal field of fractions. Let R be a Sylvester domain and let Sigma be the monoid of full matrices over R. We show that K1(U(R)) is naturally isomorphic to alpha(Sigma), the universal abelian group of Sigma. The inclusion R ⊆ U(R) induces a map K 1(R) → K1(U(R)); we also prove that if R is a fully atomic semifir (e.g. if R is a fir) then K1(U(R)) = K1(R) X D(R), where K1(R) denotes the image of K1(R) in K 1(U(R)) and D(R) is the free abelian group on the set of equivalence classes of stably associated matrix atoms over R.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics