Comment by the author, April 6, 2000: We can give an improved lower bound for the modulo m rank of co-triangle matrices.
Theorem. Let A be an n x n co-triangle matrix over a ring R, where |R|=m. Then rankR(A) >= logm(n).
Proof. Let A=BC over R, where B is an n x r and C is an r x n matrix. Since every column of A is different, all columns of C should also be different. The entries of C are chosen from the ring R of cardinality m, so there are at most mr pairwise different vectors of length r; hence the claimed inequality follows.