Title:

Universal fields of fractions : their orderings and determinants

We are concerned with two problems. Firstly, given a ring R and an epic Rfield K, under what conditions can K be fully ordered? Epic Rfields 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 Rfield 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_{1}(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_{1}(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) → K_{1}(U(R)); we also prove that if R is a fully atomic semifir (e.g. if R is a fir) then K_{1}(U(R)) = K_{1}(R) X D(R), where K_{1}(R) denotes the image of K_{1}(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.
