Let $ X$ be a topological space. The set $C(X)$ is all continuous, real valued functions on $X$, so that $C(X) \subseteq \mathbb{R}^{X}$.
Define:
$C^* (X) = \{ f \in C(X) \vert \quad f\quad is \quad bounded \}$.
My questions are:
1:The ring $C(\mathbb{R})$ is isomorphic with a proper subring.[Consider the function that are constant on $[ 0 , 1]$.] But $C(\mathbb{R})$ has no proper summand.
2: In $C(X)$ ( or $C^{*} (X)$ ), all positive units have the same number of square roots.