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Suppose A and B are square matrices of the same size over a PID R. Does the Smith Normal form of AB equal the product of the Smith normal form of A and B? I think this should be false. However, I can't quite find an example.

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    $\begingroup$ It is false. Try $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$. Their Smith normal forms are both $A$, but their product's Smith normal form is not $A^2$. $\endgroup$ – darij grinberg Dec 2 '18 at 18:40
  • $\begingroup$ Wow! Thanks a lot. $\endgroup$ – justanothermathstudent Dec 2 '18 at 19:28

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