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Mathematische Zeitschrift

, Volume 237, Issue 2, pp 259–273 | Cite as

A representation theorem for certain partially ordered commutative rings

  • Thomas Jacobi
Original article

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.)

Keywords

Continuous Function Commutative Ring Representation Theorem Weak Topology Hausdorff Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Thomas Jacobi
    • 1
  1. 1.Gorch-Fock-Str. 35, 70619 Stuttgart, Germany (e-mail: jacobi@fmi.uni-konstanz.de) DE

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