Can somebody shed some light on what is so important about these zero columns? I would think that the number of independent columns is important. I just do not see why zero columns are necessary at all.

Isn't the ideal generated by the following matrix, without any zero columns, also a nontrivial left ideal:

I think you can show that for any left ideal I there is a unique vector subspace L of [tex]\mathbf{R}^n[/tex] such that [tex]I=\{A\in I:\,AL=0\}[/tex]. Then, for any k-dimensional subspace of [tex]\mathbf{R}^n[/tex] you can choose a basis such that the first k vectors are in L. This will put your ideal into a canonical form that you are looking form .

Yes, I see that there is a subspace of [tex]\mathbf{R}^n[/tex] that is contained in the kernel of every M ∈ I. This subspace is the orthogonal complement of the row space of matrix A ∈ I that has maximum number of independent rows.

Ah yes I guess that was the idea: on an appropriate basis there will be zero columns.