On nest modules of matrices over division rings

Let m,ninmathbbN m,ninmathbbN, D D be a division ring, and M mtimesn (D) Mmtimesn(D) denote the bimodule of all mtimesn mtimesn matrices with entries from D D. First, we characterize one-sided submodules of M mtimesn (D) Mmtimesn(D) in terms of left row reduced echelon or right column reduced echelon matrices with entries from D D. Next, we introduce the notion of a nest module of matrices with entries from D D. We then characterize submodules of nest modules of matrices over D D in terms of certain finite sequences of left row reduced echelon or right column reduced echelon matrices with entries from D 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.