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Transactions of the American Mathematical Society

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

 

 

Embeddings in division rings


Author: John Dauns
Journal: Trans. Amer. Math. Soc. 150 (1970), 287-299
MSC: Primary 16.46
DOI: https://doi.org/10.1090/S0002-9947-1970-0262291-5
MathSciNet review: 0262291
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Abstract: A method for embedding a certain class of integral domains in division rings is devised. Integral domains $ A$ are constructed with a generalized valuation into a (noncommutative) totally ordered semigroup that need not be discrete. Then the multiplicative semigroup $ A\backslash \{ 0\} $ is expressed as an inverse limit of semigroups each of which is embeddable in a group. Thus $ A\backslash \{ 0\} $ can be embedded in a group $ G$. The main problem is to introduce addition on $ G$ in order that $ G$ becomes a division ring by the use of eventually commuting maps of inverse limits.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0262291-5
Keywords: Integral domain, division ring, generalized valuation into a semigroup, Ore condition, eventually commuting maps of inverse limits, P. M. Cohn's embedding theorem, totally ordered cancellative semigroup, semigroup congruence
Article copyright: © Copyright 1970 American Mathematical Society