Vol. 24, No. 1, 1968

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Second note on David Harrison’s theory of preprimes

D. W. Dubois

Vol. 24 (1968), No. 1, 57–68

Some classes of partially ordered rings are studied by means of representations in C(X), where X is a compact Hausdorff space. The first theorem generalizes the characterizations, due, respectively, to Harrison and the author, of the subrings of the real field, and the subrings of C(X). Among the many consequences proved are the following:

1. If A is a simple ring (no two-sided ideals) partially ordered so that (a) if n is a positive integer and nαj 0, then x 0; and (b) for all x in A there exists an integer n exceeding x; then A is a commutative field.

2. Let F be a field, let P be a conic prime in F whose intersection Pc in the center C of F is a primitive (AC) cone in C. Then Pc is an Archimedean order in C.

3. The compact Hausdorff space X admits a base of power at most 20 of open-and-closed sets if and only if C(X) contains a dense subfield.

Mathematical Subject Classification
Primary: 06.85
Secondary: 16.00
Received: 22 July 1966
Published: 1 January 1968
D. W. Dubois