TITLE: Coordinate-free definition of elementary divisors
QUESTION [6 upvotes]: There is a general mantra in math which says that what is independent of bases shall be defined independent of bases. Well, it is well known that the elementary divisors of a linear map $M\xrightarrow{\ \ f\ \ }N$ of finitely generated free modules over a principal ideal domain $R$ are independent of bases. So I wonder whether there is a simple definition, which does not mention chosen bases of $M$ and $N$.
Perhaps such a characterization would help to make the theory more streamlined. Of course it should be possible to prove that given elementary divisors $\alpha_1,\dots,\alpha_s$ of $f$ there are bases of $M$ and $N$ such that the corresponding matrix of $f$ has Smith normal form whose entries are precisely the $\alpha_i$ and conversely that given bases such that the corresponding matrix has Smith normal form, its entries are elementary divisors.

REPLY [3 votes]: Let $T$ be the torsion submodule of $N/\operatorname{im}(f)$ and let $p\in R$ be a prime.  Then the valuation of $\alpha_i$ with respect to $p$ (i.e., the number of times that $\alpha_i$ is divisible by $p$) is equal to the greatest $d>0$ such that $p^{d-1}T/p^{d}T$ has dimension $> s-i$ as a vector space over the field $R/(p)$ (or $0$ if no such $d$ exists).  This is easy to see from the fact that $T\cong\bigoplus_i R/(\alpha_i)$ and $\alpha_i\mid \alpha_{i+1}$ for all $i$.  Since the divisors $\alpha_i$ are only defined up to units, this completely determines them.