TITLE: Finding an algebra of smooth functions on a manifold with a given product.
QUESTION [6 upvotes]: I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of problem. I have had some exposure to elementary differential geometry (especially of curves and surfaces) but I do not know how I should think about this exercise: is there a standard method to approach it or should I just know the answer from other sources (e.g classical examples of differential geometry)?
Could you please point me in the right direction?

Suppose that the $\mathbb{R}$-algebras $\mathcal{F}_1$ and
  $\mathcal{F}_2$, as vector spaces, are isomorphic to the plane
  $\mathbb{R}^2$. Let the multiplication in $\mathcal{F}_1$ and
  $\mathcal{F}_2$ be respectively given by the relations
$(x_1, y_1) \cdot (x_2, y_2) = (x_1x_2, y_1y_2) \quad (1)$
$(x_1, y_1) \cdot (x_2, y_2) = (x_1x_2 + y_1y_2, x_1y_2 + x_2y_1) \quad (2)$
Find the manifold $M_i$ for which the algebra  $\mathcal{F}_i$, $i =1, 2$, is the algebra of smooth functions, explicitly indicating what
  function on $M_i$ corresponds to the element $(x, y) \in \mathcal{F}_i$. Are the algebras $\mathcal{F}_1$ and $\mathcal{F}_2$ isomorphic?

REPLY [5 votes]: It is immediate that $F_1 = \mathbb{R} \times \mathbb{R} \cong C^{\infty}(1+1)$, where $1+1$ is the discrete manifold with two elements. The second algebra has the unit $(1,0)$, and it is generated by $u=(0,1)$, satisfying $u^2=1$. Hence, $F_2 \cong \mathbb{R}[x]/(u^2-1) \cong \mathbb{R}[u]/(u+1) \times \mathbb{R}[u]/(u-1) \cong \mathbb{R} \times \mathbb{R} = F_1$. It maps
$$F_2 \ni (x,y) \mapsto \overline{x+uy} \mapsto (\overline{x+uy},\overline{x+uy}) \mapsto (x-y,x+y) \in F_1.$$