On modules of linear transformations

TL;DR

This paper investigates the structure of submodules of the space of linear transformations between vector spaces over a division ring, establishing conditions under which these submodules are finitely generated based on their images or coimages.

Contribution

It provides foundational results on submodules of linear transformation modules, particularly relating finite-dimensional images or coimages to finite generation.

Findings

Right submodules are finitely generated if their images are finite-dimensional.

Left submodules are finitely generated if their coimages are finite-dimensional.

Basic properties of submodules of linear transformation spaces are established.

Abstract

Let $D$ be a division ring, $\mathcal V$ and $ \mathcal W$ vector spaces over $D$, and ${\mathcal L(\mathcal V,\mathcal W)}$ the ${\mathcal L(\mathcal W)}$-${\mathcal L(\mathcal V)}$ bimodule of all linear transformations from $\mathcal V$ into $\mathcal W$. We prove some basic results about certain submodules of $\mathcal L(\mathcal V, \mathcal W)$. For instance, we show, among other results, that a right submodule (resp. left submodule) of ${\mathcal L(\mathcal V,\mathcal W)}$ is finitely generated whenever its image (resp. coimage) is finite-dimensional.