Is the following statement true: "Every smooth manifold $M$, which is a ring in the category of manifolds, must be diffeomorphic to $\mathbb{R}^n$."? (Actually, homeomorphic would suffice.) I assume manifolds to be Hausdorff, second-countable and positive-dimensional, to exclude finite rings.
I have strong feelings that this must be the case. Is there a "simple" proof of it? I know next to nothing about the theory of Lie groups, so any argument using these would have to be simple for me to understand it. On the other hand, I feel quite comfortable with "standard" algebraic topology (elementary homotopy theory, homology, cohomology rings and so on...).
Edit: I am very sorry for not making this clear the first time, but I assume all manifolds to be without boundary.