Embedding a partially ordered ring in a division algebra

Author:
William H. Reynolds

Journal:
Trans. Amer. Math. Soc. **158** (1971), 293-300

MSC:
Primary 16.80; Secondary 06.00

MathSciNet review:
0283026

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Abstract: D. K. Harrison has shown that if a ring with identity has a positive cone that is an infinite prime (a subsemiring that contains 1 and is maximal with respect to avoiding -- 1), and if the cone satisfies a certain archimedean condition for all elements of the ring, then there exists an order isomorphism of the ring into the real field. Our main result shows that if Harrison's archimedean condition is weakened so as to apply only to the elements of the cone and if a certain centrality relation is satisfied, then there exists an order isomorphism of the ring into a division algebra that is algebraic over a subfield of the real field. Also, Harrison's result and a related theorem of D. W. Dubois are extended to rings without identity; in so doing, it is shown that order isomorphic subrings of the real field are identical.

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DOI:
https://doi.org/10.1090/S0002-9947-1971-0283026-7

Keywords:
Harrison infinite prime,
ordered ring,
representation,
real-valued continuous function,
compact space,
algebraic division algebra

Article copyright:
© Copyright 1971
American Mathematical Society