# Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem

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?

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