Embeddings in division rings

Dauns, John

doi:10.1090/S0002-9947-1970-0262291-5

Embeddings in division rings
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by John Dauns

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

DOI: https://doi.org/10.1090/S0002-9947-1970-0262291-5

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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.

References

A. J. Bowtell, On a question of Mal′cev, J. Algebra 7 (1967), 126–139. MR 230750, DOI 10.1016/0021-8693(67)90071-3
P. M. Cohn, On the embedding of rings in skew fields, Proc. London Math. Soc. (3) 11 (1961), 511–530. MR 136632, DOI 10.1112/plms/s3-11.1.511
Paul Conrad and John Dauns, An embedding theorem for lattice-ordered fields, Pacific J. Math. 30 (1969), 385–398. MR 246859
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
Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
Abraham A. Klein, Rings nonembeddable in fields with multiplicative semi-groups embeddable in groups, J. Algebra 7 (1967), 100–125. MR 230749, DOI 10.1016/0021-8693(67)90070-1