Let $R$ be a full-rank $n \times m$ rectangular matrix with entries in a DVR, with $n \leq m$ (less rows than columns) and we may assume $n < m$ if useful. Is there an $m \times m$ invertible matrix $A$ in the same field such that $RA$ has at most one nonzero element on each column?
Note that this is like asking for (little less than) half of the conclusion of Smith Normal Form with (little more than) half of the assumptions.