Embedding rings with a maximal cone and rings with an involution in quaternion algebras

Authors:
Carl W. Kohls and William H. Reynolds

Journal:
Trans. Amer. Math. Soc. **176** (1973), 411-419

MSC:
Primary 16A28; Secondary 06A70, 46K99

DOI:
https://doi.org/10.1090/S0002-9947-1973-0313302-2

MathSciNet review:
0313302

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Abstract: Sufficient conditions are given for an algebra over a totally ordered field *F* to be isomorphic to a subring of the algebra of quaternions over the real closure of *F*. These conditions include either the requirement that the nonnegative scalars form a maximal cone in the algebra, or that the algebra have an involution such that the scalars are the only symmetric elements. For many matrix algebras, the cone requirement alone is imposed.

**[1]**D. K. Harrison,*Finite and infinite primes for rings and fields*, Mem. Amer. Math. Soc. No.**68**(1966), 62. MR**0207735****[2]**I. N. Herstein and Susan Montgomery,*A note on division rings with involutions*, Michigan Math. J.**18**(1971), 75–79. MR**0283017****[3]**D. G. Northcott,*An introduction to homological algebra*, Cambridge University Press, New York, 1960. MR**0118752****[4]**R. S. Palais,*The classification of real division algebras*, Amer. Math. Monthly**75**(1968), 366–368. MR**0228539**, https://doi.org/10.2307/2313414**[5]**William H. Reynolds,*Embedding a partially ordered ring in a division algebra*, Trans. Amer. Math. Soc.**158**(1971), 293–300. MR**0283026**, https://doi.org/10.1090/S0002-9947-1971-0283026-7

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0313302-2

Article copyright:
© Copyright 1973
American Mathematical Society