(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the precise question in terms of commutative algebra.)

Let ${\cal A}$ be the category of finitely generated (associative unital) commutative ${\mathbb{C}}$-algebras with exactly two idempotents.

Let ${f, g : A \rightarrow \mathbb{C}}$ be two maps in ${\cal A}$.

Does it exist a map ${c : A \rightarrow C}$ (always in ${\cal A}$) with $C$ a curve such that both $f$ and $g$ factor through $c$? (I.e. such that there are maps ${f', g' : C \rightarrow \mathbb{C}}$ such that ${f' c = f}$ and ${g' c = g}$.)

Here by "curve" I mean a connected finitely generated $\mathbb{C}$-algebra of Krull dimension 1.