Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 0262291
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] A. J. Bowtell, On a question of Mal′cev, J. Algebra 7 (1967), 126–139. MR 0230750
  • [2] P. M. Cohn, On the embedding of rings in skew fields, Proc. London Math. Soc. (3) 11 (1961), 511–530. MR 0136632
  • [3] Paul Conrad and John Dauns, An embedding theorem for lattice-ordered fields, Pacific J. Math. 30 (1969), 385–398. MR 0246859
  • [4] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864
  • [5] Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR 0222106
  • [6] Abraham A. Klein, Rings nonembeddable in fields with multiplicative semi-groups embeddable in groups, J. Algebra 7 (1967), 100–125. MR 0230749

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16.46

Retrieve articles in all journals with MSC: 16.46

Additional Information

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