Showing a set is a basis of a vector space

Question

Let $A$ be a basis for $F$ a free abelian group. Prove that $\bar A = \{ a + pF : a \in A\}$ is a basis for $F/pF$ over the field $\Bbb Z/p \Bbb Z$.

If we define scalar multiplication from $\Bbb Z/p \Bbb Z \times F/pF \rightarrow \ F/pF$ as $$\bar c \bar v = (c + p \Bbb Z)(v+pF)=(cv + pF)$$ I see that I would have to show that $\bar A$ is linearly independent and spans $F/pF$.

This means showing $$\sum_i \bar c_i \bar a_i = \bar 0 \Rightarrow \bar c_i=\bar 0 \text{ or }\sum_i c_ia_i = pf \Rightarrow c_i=pk$$ for some $k \in \Bbb Z$.

Additionally, for the spanning condition I would need to show that $x \in F/pF \Rightarrow x \in span(\bar A)$, or $x = h + pF \Rightarrow x = \sum_ic_ia_i + pF$ for some $c_i \in Z$ and $a_i \in F$

But I'm stuck at both of these points. Anyone have any ideas?

Answer 1

Hint:
Every $f\in F$ can be uniquely written as $\displaystyle\sum_{i\le n}\varphi_ia_i$ for some $n\in\Bbb N$, $\,\varphi_i\in\Bbb Z$ and $a_i\in A$.