Rank of homology basis in Ahlfors' Complex Analysis In Ahlfors' Complex Analysis book Section 4.4.7, he decomposes the complement in the extended plane of a region $\Omega$ into connected components. He then constructs a collection of cycles $\gamma_i$ in $\Omega$, one for each component, such that every cycle $\gamma$ in $\Omega$ is homologous to an integer linear combination of the $\gamma_i$.
Each homology class for $\Omega$ is represented uniquely as such an integer linear combination. He then mentions that there are other such collections of cycles that also serve as a homology basis, but that

...by an elementary theorem in linear algebra we may conclude that every homology basis has the same number of elements.

I think I understand at least partially why this should be true: Homology classes of cycles in $\Omega$ form a module over the integers, and because we've already found the basis $\gamma_i$ we know that it's in fact a free module on these generators. We know from slightly more than elementary algebra that the rank of a finitely generated free module over a commutative ring is well-defined.
Question: What is the elementary linear algebra argument that he's using?
EDIT: Does the fact that it's a module over the integers help out? Can you embed the module of homology classes into its module of fractions (without using a basis to do so), show that the homology basis for the module becomes a basis for the module of fractions as a vector space over the rationals. Then you could use the linear algebra result for dimension of a vector space?
Thank you!
 A: Concerning your edit, yes, you can embed the integer homology group into the homology with rational coefficients. You can define chains with rational coefficients for the paths, a cycle is a chain with empty boundary (the boundary of a path is $\text{endpoint} - \text{startpoint}$, the boundary of a chain is the corresponding linear combination of the boundaries of the constituent paths). The integral of $f$ over $\Gamma = \sum c_i\gamma_i$ is
$$\sum c_i\cdot \int_{\gamma_i} f(z)\,dz,$$
the winding number of a chain around a point $a$ not on the trace of the chain is
$$n(\Gamma,a) = \frac{1}{2\pi i}\sum c_i\int_{\gamma_i} \frac{dz}{z-a},$$
two cycles are homologous if $n(\Gamma_1,a) = n(\Gamma_2,a)$ for all $a\notin \Omega$.
A more or less tedious verification shows that $[\Gamma]_\mathbb{Z} \mapsto [\Gamma]_\mathbb{Q}$ is a well-defined embedding of the integer homology into the rational homology.
But determining the rank of a finitely generated free abelian group $B$ by counting the elements of the $\mathbb{F}_p$-vector space $B/pB$ for a prime $p$ ($p = 2$ for example) seems easier and more elementary to me.
