A representation theorem for certain partially ordered commutative rings

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.)