I'm studying modules by reading Dummit and Foote, and I'm having a problem understanding the definition of a free module. I read this stackexchange question, but I couldn't figure it out.
The textbook defines a free module as following:
The following is the example that I'm confused about.
Let $F = \{ 0 \}$. Then $F$ is a $\mathbb{Z}$-module since $F$ is an abelian group under addition.
In the stackexchange question, the empty set is given as the basis for $F$. That makes sense because $F$ has no nonzero elements, so it satisfies the definition vacuously.
But, using the same logic, wouldn't $F$ be a free $\mathbb{Z}$-module on the set $\{ 0 \}$?
I think that would be a problem because if that was the case, the rank of $\{ 0 \}$ would be both 0 and 1.
The set $\{0\}$ is not linearly independent, because $r0=0$ for any nonzero $r\neq 0$.
The empty set does span the zero module, because an empty sum is zero.