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 K<sub>1</sub>(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 K<sub>1</sub>(U(R)) is naturally isomorphic to alpha(Sigma), the universal abelian group of Sigma. The inclusion R ⊆ U(R) induces a map K<sub> 1</sub>(R) → K<sub>1</sub>(U(R)); we also prove that if R is a fully atomic semifir (e.g. if R is a fir) then K<sub>1</sub>(U(R)) = K<sub>1</sub>(R) X D(R), where K<sub>1</sub>(R) denotes the image of K<sub>1</sub>(R) in K<sub> 1</sub>(U(R)) and D(R) is the free abelian group on the set of equivalence classes of stably associated matrix atoms over R.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:704479 |
| Date | January 1981 |
| Creators | Revesz, Gabor |
| Publisher | Royal Holloway, University of London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://repository.royalholloway.ac.uk/items/070472ef-1f40-430b-ac17-2afafd76e88e/1/ |
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