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

TL;DR

This paper classifies nonsplit module extensions over a specific algebra generated by two elements with a relation, revealing conditions under which such extensions occur, including dimensionality constraints of the modules involved.

Contribution

It provides a complete description of simple modules over the algebra and characterizes when nonsplit extensions exist, extending understanding of module theory over this algebra.

Findings

Nonsplit extensions only occur between one-dimensional modules or under specific conditions involving infinite-dimensional modules.

The structure of simple modules over the algebra is fully described.

Conditions for the existence of nonsplit extensions are explicitly characterized.

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 $0\rightarrow U\rightarrow E \rightarrow V\rightarrow 0$, 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.