  Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned
into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \times p$ submatrix of $M$
with columns from distinct blocks $M_i$ is invertible, then there is an
invertible $p \times p$ matrix $Q$ and a positive integer $m \leq p$ such that
$QM = [QM_1, ..., QM_n]$ is in reduced echelon form and in all but at most
$m-1$ blocks $QM_i$ the last $m$ entries of each column are either all zero or
they include a non-zero non-unit.
