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!

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.
