, Volume 237, Issue 2, pp 259-273

A representation theorem for certain partially ordered commutative rings

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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

\[ \Phi_M^{-1}({\cal C}^+(X(M),\R))\!=\!\{a\in A\mid \forall\,n\in\N\; \exists\,k\in\N\setminus\{0\}\; k(1+ na)\in M\} \]

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.)

Received March 4, 1999; in final form January 15, 2000 / Published online February 5, 2001