Embeddings in division rings

Author:
John Dauns

Journal:
Trans. Amer. Math. Soc. **150** (1970), 287-299

MSC:
Primary 16.46

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 are constructed with a generalized valuation into a (noncommutative) totally ordered semigroup that need not be discrete. Then the multiplicative semigroup is expressed as an inverse limit of semigroups each of which is embeddable in a group. Thus can be embedded in a group . The main problem is to introduce addition on in order that becomes a division ring by the use of eventually commuting maps of inverse limits.

**[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**

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DOI:
http://dx.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