Rings of almost everywhere defined functions

We prove the following representation theorem: A partially ordered commutative ring R is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space X if and only if R is archimedean and localizable. Here we assume that the positive cone of R is closed under multiplication and stable under multiplication with squares, but we also show that one of these assumptions implies the other. An almost everywhere defined function on X is one that is defined on a dense open subset of X. These functions can be added and multiplied pointwise so that the result is again almost everywhere defined. A partially ordered commutative ring R is archimedean if the underlying additive partially ordered abelian group is archimedean, and R is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As application we obtain several more specific representation theorems: representations by continuous real-valued functions on some topological space if R is -bounded, and representation of lattice-ordered commutative rings (f-rings), of partially ordered fields, and of commutative operator algebras.