#### Vol. 12, No. 8, 2019

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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
##### Abstract

Let $R$ be the associative $k$-algebra generated by two elements $x$ and $y$ with defining relation $yx=1$. 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.

##### Keywords
simple modules, representations, module extensions
Primary: 16D60
Secondary: 16G99