Representation of abelian groups and rings by families of real-valued functions

Author:
Carl W. Kohls

Journal:
Proc. Amer. Math. Soc. **25** (1970), 86-92

MSC:
Primary 06.78; Secondary 46.00

MathSciNet review:
0256964

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Abstract: Sufficient conditions are indicated for the existence of an isomorphism of an abelian group or ring into the family of real-valued continuous functions on a realcompact or compact space. The spaces consist of sets of subsemigroups or subsemirings that are maximal among those containing a fixed element but not containing its additive inverse; in the ring case, where the element is the multiplicative identity, these are just infinite primes in the sense of Harrison. Earlier results of Harrison, Kadison, and Krivine follow from the present discussion.

**[1]**D. W. Dubois,*A note on David Harrison’s theory of preprimes*, Pacific J. Math.**21**(1967), 15–19. MR**0209200****[2]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[3]**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****[4]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199****[5]**D. K. Harrison,*Finite and infinite primes for rings and fields*, Mem. Amer. Math. Soc. No.**68**(1966), 62. MR**0207735****[6]**Richard V. Kadison,*A representation theory for commutative topological algebra*, Mem. Amer. Math. Soc.,**No. 7**(1951), 39. MR**0044040****[7]**J.-L. Krivine,*Anneaux préordonnés*, J. Analyse Math.**12**(1964), 307–326 (French). MR**0175937**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1970-0256964-3

Keywords:
Ordered abelian group,
ordered vector space,
ordered ring,
ordered algebra,
Archimedean totally ordered group,
representation,
real-valued continuous function,
realcompact space,
compact space,
Harrison infinite prime,
extreme point

Article copyright:
© Copyright 1970
American Mathematical Society