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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Embedding a partially ordered ring in a division algebra
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by William H. Reynolds PDF
Trans. Amer. Math. Soc. 158 (1971), 293-300 Request permission

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.
References
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 158 (1971), 293-300
  • MSC: Primary 16.80; Secondary 06.00
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0283026-7
  • MathSciNet review: 0283026