On nest modules of matrices over division rings

Abstract

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.

Will be published in the special issue of BIMS in honor of Professor Heydar Radjavi's 80th birthday