For any matrix $A$ with entries in a PID, there exist invertible matrices $P$ and $Q$ such that $B = PAQ$, where $B$ is in Smith normal form. This theorem is usually proved by using elementary row/column operations. However, in the case where the entries of $A$ are in a field, there is a short proof which is basically just the rank-nullity theorem [interpret $A$ as a linear map from $\mathbb{F}^m$ to $\mathbb{F}^n$, choose a basis $\{v_{1}, \ldots , v_{k}\}$ for the kernel of $A$ and extend it to a basis for $\mathbb{F}^m$, and finally use $\{A(v_{1}), \ldots , A(v_{k})\}$ as a basis for $\mathbb{F}^n$ (extending it as necessary)]. Of course, the rank-nullity theorem does not hold over rings, but is there some way to generalize the ideas of this proof for any PID, perhaps by passing to its field of fractions?
Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem
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ring-theory
modules