On modules of linear transformations

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.