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.

11 pages