on kernels with integers as codomain and not a field, using Milnor's bilinear function

The problem of generalizing Kernels as functions with integers and not reals as codomain is in Mercer's theorem. The problem is that the theorem begins assuming that the Kernel function has the reals and complexes as codomain. Using Milnor's definition of a bilinear function that becomes an inner product space, and the non-degenerative restriction, I can construct a Kernel-like function that works with a free module instead of a vector space. But then, how can I be certain that it will have all the properties expected of a Kernel function for SVMs?