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

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 → U → E → V → 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.2010 Mathematics Subject Classification. Primary: 16D60. Secondary: 16G99.

Injective modules over the Jacobson algebra $K< X,Y | XY = 1 >$

“…Gerritzen [11] (2000); Bavula [7] (2010); Ara and Rangaswamy [6] (2014); Iovanov and Sistko [12] (2017); and Lu, Wang and Wang [15] (2019).…”