Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, \ldots, z_n]$.  Define $R[X] = R[z_1, \ldots, z_n] / (f_1, \ldots, f_m )$ for any subring $R \subseteq \mathbb{C}$ containing $\mathbb{Z}$.

 Now, if $f\in \mathbb{C}[X]$ can be represented by a polynomial in with integer coefficients, then $\mathbb{C}[X]$ (resp. $R[X]$) can be regarded as a module over $\mathbb{C}[f]$ (resp. $R[f]$), where $\mathbb{C}[f]$ is the $\mathbb{C}$-algebra generated by $f$ in $\mathbb{C}[X]$.  Note that $\mathbb{C}[f]$ is the image of the induced homomorphims $f^\ast \colon \mathbb{C}[\mathbb{A}^1] \to \mathbb{C}[X]$.   

Suppose $\mathbb{C}[X]$ and $\mathbb{Z}[X]$ are finite over $\mathbb{C}[f]$ and $\mathbb{Z}[f]$.  For a ring $R$ as above, let $\mu_R(f)$ denote the minimal number of generators for $R[X]$ as a module over $R[f]$.  Then 
$$\mu_\mathbb{Z}(f) \geq \mu_\mathbb{Q} ( f) = \mathrm{rank}_{\mathbb{Q}[f]} \mathbb{Q}[X]$$ 
where the equality follows from the fact that $\mathbb{Q}[X]$ is a finitely generated torsion free module over a PID.  The inequality can be strict, however.    

What is the appropriate geometric interpretation of the case when $\mu_\mathbb{Z} ( f) > \mu_\mathbb{Q} (f)$?  What does this say about $f$ when thinking of it as a regular function on $X$?  Does this inequality tell us anything geometric about $X$?

From a few Overflow posts (like [this][1] and [this][2]), Corollary A3.3 in Eisenbud's *Commutative Algebra*, and glancing through Hartshorne,  it appears that this has to do with algebraic vector bundles on $\mathbb{A}^1$ and/or coherent sheaves, but I can't figure out how the regular function $f$ fits into the picture or what intersecting down to $\mathbb{Z}$ does specifically.

Edit: I believe the module structure of $\mathbb{C}[X]$ over $\mathbb{C}[f]$ corresponds to the push-forward of the trivial line bundle over $X$. This gives a vector bundle of rank $\deg f$ over $\mathbb{A}^1$. Is this correct?

Note: this is my first Overflow question and I previously asked this question on [StackExchange](http://math.stackexchange.com/q/1706695/275190) where it was upvoted a few times, but had no responses.  If it is an inappropriate question for this site, please let me know and I will delete it.    


  [1]: http://mathoverflow.net/questions/29993/rank-of-a-module
  [2]: http://mathoverflow.net/questions/61678/what-is-the-general-geometric-interpretation-of-modules-in-algebraic-geometry