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Crossposted to MathOverflow here.

Let $R$ be a principal ideal ring. If $A$ is any $p \times q$ matrix over $R$, then does there exist an invertible matrix $U$ in $\text{M}_p(R)$ such that the $p \times q$ matrix $UA$ is upper triangular?

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    $\begingroup$ What is a unit matrix? $\endgroup$ – Rene Schipperus Sep 30 '16 at 2:49
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    $\begingroup$ Did you not mean a matrix which is a unit in $M_p(R)$? (A matrix with an inverse in $M_P(R)$) $\endgroup$ – Nick R Sep 30 '16 at 3:09
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Of course you can try row reduction: you should probably go look at Smith normal form. Of course that is written for principal ideal domains, but I don't remember the domain part being the important condition to execute row operations. I think you just need GCDs. (I should check, maybe there is something tricky.)

Part of the algorithm is the process of performing row operations to put the matrix in upper triangular form.

The process continues with aim invertible matrix on the right to get the matrix into a diagonal form.

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