# Principal ideal ring, does there exist a unit matrix such that certain matrix is upper triangular?

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?

• What is a unit matrix? – Rene Schipperus Sep 30 '16 at 2:49
• Did you not mean a matrix which is a unit in $M_p(R)$? (A matrix with an inverse in $M_P(R)$) – Nick R Sep 30 '16 at 3:09