Nonsplit module extensions over the one-sided inverse of k[x]

Lu, Zheping; Wang, Linhong; Wang, Xingting

doi:10.2140/involve.2019.12.1369

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Nonsplit module extensions over the one-sided inverse of $k[x]$

Zheping Lu, Linhong Wang and Xingting Wang

Vol. 12 (2019), No. 8, 1369–1377

DOI: 10.2140/involve.2019.12.1369

Abstract

Let $R$ be the associative $k$ -algebra generated by two elements $x$ and $y$ with defining relation $y x = 1$ . A complete description of simple modules over $R$ is obtained by using the results of Irving and Gerritzen. We examine the short exact sequence $0 \to U \to E \to V \to 0$ , where $U$ and $V$ are simple $R$ -modules. It shows that nonsplit extension only occurs when both $U$ and $V$ are one-dimensional, or, under certain condition, $U$ is infinite-dimensional and $V$ is one-dimensional.

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Keywords

simple modules, representations, module extensions

Mathematical Subject Classification 2010

Primary: 16D60

Secondary: 16G99

Milestones

Received: 8 May 2019

Revised: 8 September 2019

Accepted: 9 September 2019

Published: 25 October 2019

Communicated by Scott T. Chapman

Authors

Zheping Lu
Tandon School of Engineering New York University Brooklyn, NY United States

Linhong Wang
Department of Mathematics University of Pittsburgh Pittsburgh, PA United States

Xingting Wang
Department of Mathematics Howard University Washington, DC United States

Vol. 12, No. 8, 2019