Are the notions of rank and determinantal rank equivalent for an $m\times n$ matrix $A$ with entries in a principal ideal domain $D$? I'm specifically interested in the case $D=\mathbb{Z}$.
2026-03-27 23:48:48.1774655328
Are rank and determinantal rank the same over a PID?
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If by the rank of $A$ you mean the rank of the image of (the linear map associated to) $A$, then they are equal. Note that the rank of $A$ is not affected by changes of the bases, so one can assume that $A$ is in its Smith Normal Form. Since the invariant factors of $A$ are computed by using the minors of $A$ the conclusion follows.