dg.differential geometry - Is there an easy way to describe the sheaf of smooth functions on a product manifold?

Let $Mfd$ be the category of smooth manifolds (over $mathbb{R},mathbb{C}$) and $LRS$ be the category of locally ringed spaces (over $mathbb{R},mathbb{C}$). Then the functor $Mfd to LRS$ is full and faithful and indeed, you may define $Mfd$ to be the full subcategory of $LRS$, which consists of those locally ringed spaces, which are locally isomorphic to $mathbb{R}^n$ together with the sheaf of smooth functions. Unfortunately, the functor $Mfd to LRS$ does not preserve products; see below.

It should be remarked that products in $LRS$ do exist (even infinite ones); take the obvious product in $RS$ and "make it local" by introducing new points, namely prime ideals in the stalks, and take the localizations of the stalks as the new stalks. Now if we take the product of two manifolds $M,N$ in $LRS$, we get as a topological space the usual product $M times N$; however, the structure sheaf consists only of those functions, which are locally of the form $(u,v) mapsto f(u) * g(v)$ for smooth functions $f,g$ defined locally on $M,N$, or sums and also quotients of these functions. These functions are also smooth with respect to the usual smooth structure on $M times N$, but the reverse is not true: For example it seems to be true that $mathbb{R}^2 to mathbb{R}, (u,v) mapsto exp(uv)$ is not such a functions (however, I don't know how to prove this).

However, I think that Stone-Weierstraß implies that these simple functions are dense within all smooth functions. Thus, we may regard $M times N$ in $Mfd$ as the "completion" of $M times N$ in $LRS$.