On modules of linear transformations

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