On nest modules of matrices over division rings

TL;DR

This paper characterizes submodules and ideals of nest modules of matrices over division rings, introducing the concept of nest modules and providing a detailed structural analysis of their submodules and subbimodules.

Contribution

It introduces nest modules of matrices over division rings and characterizes their submodules, subbimodules, and ideals, extending the understanding of matrix modules over division rings.

Findings

Characterization of one-sided submodules via echelon matrices

Introduction and structural analysis of nest modules

Description of subbimodules and ideals of nest modules

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.