Understanding a proof of the invariance of cardinality of finite bases of a free $R$-module 
Theorem. Let $R$ be a commutative ring with $1$. Any two finite bases of a free $R$-module $M$ have the same number of elements.
Proof. Let $M$ be a free module with a basis $\{x_1, x_2,\ldots, x_n\}$ and $I$ be a maximal ideal of $R$. Then $R/I = F$ is field. Since $V = M/IM$ is annihilated by $I$, $V$ is a vector space over $F$. If $[x_i] = x_i + IM$ $(1 ≤ i ≤ n)$, then $B = \{[x_1], [x_2], . . . , [x_n]\}$ is a basis of $V$ over $F$. Since any two finite bases of a vector space have the same number of elements, the theorem follows.

(See Theorem 9.6.1 of this book for detials)
I was trying to understand the above proof of the theorems but I didn't understand most of it,

*

*Why is $V$ a vector space over $F$ and under which operation?


*Why $V$ is annihilated by $I$?


*Why $B$ is a basis of $V$ over $F$ and why exactly does the invariance of cardinality of bases of $V$ imply the same for $M$?
Any help is appreciated.
 A: *

*We have $V \cong M \otimes_R R/I = M \otimes_R F$, which is naturally a $F$-vector space. Concretely, 
if $[r]_I \in F$ (with $r \in R$) and $[m]_{IM} \in V$ (with $m \in M$), then
$$[r]_I \cdot [m]_{IM} := [r \cdot m]_{IM}$$
provides a well-defined action of $F$ on $V$.

*Seen as an $R$-module, $V$ is annihilated by $I$ : pick   $[m]_{IM} \in V$ (with $m \in M$) and $i \in I$. Then $i [m]_{IM} = [im]_{IM} = [0]_{IM}$.

*Clearly $B$ generates $V$ over $F$. We check that $B$ is linearly independent over $F$. If $\sum_j [a_j] [x_j] = 0 \in V$ with $[a_j] \in F$, then $$w := \sum_j a_j x_j \in IM.$$ 
Because the $x_i$'s generate $M$ over $R$, we may write $w = \sum_l i_l x_l$ for some $i_l \in I$ (since $w$ lies in $IM$). What can you conclude, using the fact that the $x_i$'s are linearly independent over $R$?
Moreover, if you have another basis $\{y_1, ..., y_m\}$ of $M$ over $R$, then $\{ [y_1], ..., [y]_m \}$ gives, as before, a basis of $V$ over $F$. Therefore we must have $m=n$ since $V$ is a vector space over a field.
