  Let $ m , n \in \mathbb{N}$, $D$ be a division ring, and $M_{m \times n}(D)$
denote the bimodule of all $m \times n$ matrices with entries from $D$. First,
we characterize one-sided submodules of $M_{m \times n}(D)$ in terms of left
row reduced echelon or right column reduced echelon matrices with entries from
$D$. Next, we introduce the notion of a nest module of matrices with entries
from $D$. We then characterize submodules of nest modules of matrices over $D$
in terms of certain finite sequences of left row reduced echelon or right
column reduced echelon matrices with entries from $D$. We use this result to
characterize principal submodules of nest modules. We also describe
subbimodules of nest modules of matrices. As a consequence, we characterize
(one-sided) ideals of nest algebras of matrices over division rings.
