Purely R-algebraic characterization of smooth manifolds among R-algebraifolds

Develop a characterization, using only R-algebra structure, of those R-algebraifolds A that are isomorphic to C^\infty(M) for a connected smooth manifold M; i.e., provide algebraic criteria that identify C^\infty(M) within R-algebraifolds without invoking the language of C^\infty-rings.

Background

The functor sending a connected smooth manifold M to its algebra of smooth functions C^\infty(M) is fully faithful, and characterizations exist in the framework of C^\infty-rings.

The authors point out that a characterization relying solely on the structure of the underlying R-algebra (without C^\infty-ring machinery) is, to their knowledge, not available.