Mathematische Zeitschrift

, Volume 237, Issue 2, pp 259–273

A representation theorem for certain partially ordered commutative rings

Authors

  • Thomas Jacobi
    • Gorch-Fock-Str. 35, 70619 Stuttgart, Germany (e-mail: jacobi@fmi.uni-konstanz.de)
Original article

DOI: 10.1007/PL00004868

Cite this article as:
Jacobi, T. Math Z (2001) 237: 259. doi:10.1007/PL00004868

Abstract.

Let A be a commutative ring with 1, let \(P\subset A\) be a preordering of higher level (i.e. \(0,1\in P, -1\not\in P, P+P\subset P, P\cdot P\subset P\) and \(A^{2n}\subset P\) for some \(n\in\N\)) and let \(M\subset A\) be an archimedean P-module (i.e. \(1\in M, -1\not\in M, M+M\subset M, P\cdot M\subset M\) and \(\forall\,a\in A\;\exists\,n\in\N\;\; n-a\in M\)). We endow \(X(M):=\{\varphi\in{\rm Hom}(A,\R)\mid\varphi(M)\subset\R_+\}\) with the weak topology with respect to all mappings \(\widehat{a}:X(M)\rightarrow\R, \widehat{a}(\varphi):=\varphi(a)\) and consider the representation \(\Phi_M:A\rightarrow{\cal C}(X(M),\R), a\mapsto\widehat{a}\). We find that X(M) is a non-empty compact Hausdorff space. Further we prove that

\(\)

and \({\rm ker}(\Phi_M)=\{a\in A\mid \forall\,n\in\N\;\exists\,k\in\N\;\; k(1\pm na)\in M\}\). (By \({\cal C}^+(X(M),\R)\) we denote the set of all continuous functions which do not take negative values.)

Copyright information

© Springer-Verlag Berlin Heidelberg 2001