# On nest modules of matrices over division rings

Authors

1 Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan 19395-5746, Iran.

2 Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.

Abstract

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.

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### History

• Receive Date: 15 September 2014
• Revise Date: 17 January 2015
• Accept Date: 26 January 2015