On nest modules of matrices over division rings

‎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.