M. Rahimi-Alangi | Searcharxiv

arXiv2015

On modules of linear transformations

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.

math.RA↗

arXiv2015

On nest modules of matrices over division rings

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.

math.RA↗